Short Answer: H.R. 4 was opposed by Republicans supporting Rodney Davis' (R-IL) opposition to the bill in it's present form, primarily because of Republican opposition to any attempts of creating a vehicle for campaign financing subsidies from public funds. Since Democratic party members were unwilling to allow Davis' proposed remedy, the Republicans (except one) voted Nay.
In the following I detail why this is the correct answer.
First, how can one answer any question "Why do (Republicans/Democrats) oppose (some Bill)"? By supposition of motive? How do you know that the motive you guess is correct? By reading journalist's own guess work? How do you know the journalists are right? Or by trying to find the answer by listening to statements made about the subject first hand? I suggest it is the latter. Even if you like to suppose a certain motive, you would still have to demonstrate it in the arguments and have to consider the involved people's own statements before dismissing them and overriding them with your (or some journalist's) suppositions.
This particular question here is an interesting example case as there is no news article to be easily found on web search engines that gives an opposing statement. So how can we know what the motivations of the representatives were?
We have to go into the Congressional Records and read about the statements made in debate. Here is how you can do that. First you look up the bill, in this case H.R. 4 of the 116th congress 2019/20.
There in we see a plenary debate that was scheduled for one hour on December 6. So we can go to the Congressional Record and look up the House's plenary schedule on that date.
There we find the debate on H.R. 4 by searching the words, and here are the debate minutes.
It doesn't look like this debate took all of an hour that was scheduled for it, since only 2 representatives spoke about it over 5 minutes, and there were 2 recorded electronic votes that were set to take only 5 minutes.
So, it appears that the representatives present had mostly made up their mind or were behind the debate contribution of their single colleague who spoke. In case of the Republican Opposition the representative was Rodney Davis from Illinois.
When asked by the speaker pro tempore whether he opposed the bill, Davis replied he was opposed in its current form, and then proceeded to propose an amendment which, we may assume might have been at least a partial remedy to his opposition, and by the fact that no other Republican representative spoke on the subject, one may further assume that Davis was leading the Republican vote to a large extent. Here is this exchange from the record and proposed amendment:
Mr. RODNEY DAVIS of Illinois. Mr. Speaker, I have a motion to
recommit at the desk. The SPEAKER pro tempore. Is the gentleman
opposed to the bill? Mr. RODNEY DAVIS of Illinois. I am in its
current form. The SPEAKER pro tempore. The Clerk will report the
motion to recommit. The Clerk read as follows:
Mr. Davis moves to recommit the bill H.R. 4 to the Committee on
the Judiciary with instructions to report the same back to the House
forthwith with the following amendment: Page 39, after line 9,
insert the following:
SEC. 11. RULE OF CONSTRUCTION.
Nothing in this Act or the amendments made by this Act may be
construed to allow fines or other amounts paid to the United States
in connection with a violation of title I of the Voting Rights Act
of 1965 (52 U.S.C. 10301 et seq.), including any amount paid
pursuant to a settlement agreement (including a plea agreement,
deferred prosecution agreement, or non-prosecution agreement), to be
used to make a payment in support of a campaign for election for the
office of Senator or Representative in, or Delegate or Resident
Commissioner to, the Congress.
In his short remarks, Davis explained that his concern was about the idea of having tax dollar supported campaign financing subsidies, along with other attempts to control elections by federal interventions. In his own words:
In October, the majority jammed through H.R. 4617, the SHIELD Act,
an attempt to federally hijack campaign finance law in this country.
In June, the majority jammed through H.R. 2722, the SAFE Act, an
attempt to federally hijack election infrastructure in this country.
And in February, the majority jammed through H.R. 1, the For the
People Act, a piece of legislation that, as introduced, would fund
all of our campaigns with tax dollars from hardworking Americans.
Catchy titles can't hide the facts, and the facts are that these
four bills are bad partisan policy that would negatively affect the
American people.
When the Democrats proposed public financing of campaigns in H.R. 1,
I could hardly believe it. The 6-to-1 small-dollar campaign match
program would create a mandatory donation from the American taxpayer
to a political candidate.
For every $200 donated by hardworking Americans to any political
campaign of all of us in this institution, the Federal Government, on
the backs of the taxpayers, would give $1,200 to that same
politician's campaign.
This program would do nothing but fill the swamp, and any Member who
voted for it was voting to fill their own pockets and the pockets of
political operatives nationwide.
At Rules Committee, though, this was changed. The shell game now
includes a fund which is supposedly financed through fines and
settlements. But we have now seen the CBO score, and this fund does
not support itself.
So what happens when it fails? I will tell you. It will ultimately
fall to the taxpayers in this country to support this Democratic
policy.
So the issue is about public funds flowing to political campaigns. This is a pretty significant issue and the issue addressed by Davis' proposed amendment.[+]
As an earlier answer by @Machavity has explained in much detail, the Voting Rights Act of 1965 is largely intact, except for the preclearance issue. As even a cursory analysis of the text of this bill H.R. 4 shows, it is mostly about this preclearance issue that was struck down by SCOTUS in Section 4 and left Section 5 unenforceable. For media soundbites and partisan propaganda people may claim that this is about voting rights, when in reality all of the provisions of the Voting Rights Act of 1965 are intact.
As a side note, the records show that the Voting Rights Act of 1965 was supported by the vast majority of Republicans and more Democrats voted Nay than Republicans even in a Democrat controlled house, in fact only 6% of Republicans voting Nay vs. 26% of Democrats voting Nay. Just to remind that civil rights issues in America were first introduced by the Republicans, including Civil Rights Act bills prior to 1964 that all failed on Democrat opposition to any Civil Rights legislation. Evaluation of the Goldwater opposition of 1964 needs to follow the same rules and example I am exemplifying here: first to take into account the statements made by the people in question, and then only consider hypotheses contrary to their own statements, where the null-hypotheses ought to be the explanation given in the people's own statements.
Therefore, the conclusion must be that H.R. 4 was opposed by Republicans supporting Rodney Davis' (R-IL) opposition to the bill in its present form, primarily because of Republican's opposition to any attempts of creating a vehicle for campaign financing subsidies from public funds. Since Democratic party members were unwilling to allow Davis' proposed remedy, the Republicans (except one) voted Nay.
There are many additional facts one might ponder in this question. In no particular order:
1. The bill was presumably discussed in more detail in the committee session(s) previously and reached the plenary with most representatives wanting to get over it and therefore aligning with one leading colleague. To find more about the discussion we may go back to the committee meeting minutes, using the same methodology and resources I have exemplified above.
2. One may wonder what the rationale of the Democratic party opposition to Davis' proposed amendment were, a good question since you recall that the amendment had nothing to do with actual voting rights rules, not even with the preclearance issue which one might well oppose for being undue federal influence and being a rule that is not equal for all counties, being possibly very messy (which is why SCOTUS struck it down in the first place.) So one might assume that if the issue was indeed protecting voting rights, a compromise on Davis' amendment should have been possible. Why were the Democratic party representatives unwilling to entertain such compromise?
Again, the only reasonable way to try to find the answer to this question is to first receive the statement by the leader of the opposition to Davis' amendment in his own words. Ms Sewell of Alabama spoke for (less than) 5 minutes. In summary first, having read Sewell's statement several times, I cannot find an actual rationale why the prohibition of use of fines for public campaign financing is detrimental to voting rights, except perhaps for the beginning, which is:
Ms. SEWELL of Alabama. Mr. Speaker, I oppose this amendment because
it is a mere distraction. It is an attempt to politicize the Voting
Rights Act of 1965 by interjecting campaign finance and settlement
terms into civil rights legislation.
If Republicans were really serious about voting rights--about voting
rights--they would actually be willing to come to the table and talk
about how we can fully restore section 4 of the Voting Rights Act of
1965.
Leave it up to our colleagues across the aisle to interject money
and finance into civil rights law. What has been lost today in this
debate is the very heart of this bill; it is the central meaning of
the bill.
All we have is a very general accusation that a prohibition to use fines from the Voting Rights Act for campaign finance subsidies would be "an attempt to politicize the Voting Rights Act". I believe it is fair to say that when any representative accuses another of "politicizing" but then goes ahead and delivers no actual argument against the point made, instead meanders into invoking names of heroes and shouting "Shame on you!" there is at least a doubt left behind that such pathos might be uttered in an attempt to hide ulterior motives which may in its own right be called "political".
3. A third such "additional fact" in answering the question why Republicans might have voted on party line against the bill is to consider what sort of House votes on any resolution were achieved in the 116th Congress that are not on party line? I suspect the country, and hence the Congress, is so divided that there is simply no care for coming up with motions palatable to the other party's representatives.
Postscriptum
I am making a good faith effort in trying a fair unbiased assessment of such a "Why" question by demonstrating a method of fact finding and interpretation. It is probably easy to guess what my own position on the matter is, so I do not claim that I am unbiased. What I do claim is that I am making an effort to base my answer in specific facts of record, and not innuendo, and I demonstrated my fact finding and interpretation methods on two opposing statements.
I promise I will edit my answer to account for any other records of statements made by Republicans which may shed additional or different light on the answer I found so far. Please bring such facts up in the comment and I will address them by editing my answer.
UPDATES: (Rather than including updates into my lengthy text above, I will for now add them at the end. A complete rewrite would include it where it belongs, but to be fair with people who might already have done the work of digesting my answer I put it here in the form of footnotes.
[+] I do not want to make it sound as if my conclusion should be so narrowly drawn as if the federal funding of campaigns would be the only issue predicating the overall Republican sentiment behind their rejection of the bill. There is more contained in Davis' rejection speech. Davis' had mentioned dissatisfaction about several related bills that he claims the Democratic Party representatives had "jammed though" previously, and those other concerns are about federal control of elections. This brings up two thoughts:
- Very likely (as mentioned as additional consideration point 3. above) theirs is a reaction to the sense of present House Majority just "jamming through" Bills without taking objections from the minority seriously. Obviously if railroading is your leadership style you will not get bi-partisan support. This is a self-fulfilling prophesy and works both ways: you assume bad faith (as the most popular answer to date here does), you think the minority has no valid concern to contribute, you relish in the fact that you have the majority, and therefore you won't need their support, you therefore have a good chance of not getting minority support.
- In this concern about "federal control" one might impute the phrase "states rights" into Davis' speech, or believe someone else aligned with him might have used it. This has come up in some of the other answers and comments. What I find surprising about this is why, if Republicans were focused so purely on garnering the upper hand in elections, why would they not make use of the current executive power to go along with the proposal and use similarly arbitrary ruling as the Holder DoJ (that lead to the SCOTUS strike-down)? Federal control goes both ways. There is a great interest in Federal control of voter fraud right now, and hence, if the Republicans were not sincere with the concern about "federal control", you would expect them to be vying for using that control.
The point is: to arrive at an unbiased answer we must base our answer on the words and actions of the subject of our investigation (rather than the re-framing of their opponents). But we should not need to use tunnel vision, sticking only to the most narrow answer. The Davis' amendment was probably a necessary condition to get some republicans to vote in favor of the bill, but we can't be sure that it would have been a sufficient condition to garner significantly more of their support. One may even claim that it is only a tactical proposal that was so squarely unpalatable to the majority, that it was designed to keep most in the minority to keep opposing. But if that should be our hypothesis in this case, it also would put in doubt the true motivations of the majority in proposing this bill and rejecting the amendment that seems to have so little to do with the purported subject of the bill. In that case, it might actually give weight to Davis' amendment as a genuine matter, not just a tactical one.