Congressional Research Service (2015): Impeachment and Removal (PDF)
Judgment of the Senate
When the presentation of evidence and argument by the managers and counsel for the respondent has concluded, the Senate as a whole meets in closed session to deliberate. Voting on whether to convict on the articles of impeachment commences upon return to open session, with yeas and nays being tallied as to each article separately. A conviction on an article of impeachment requires a two-thirds vote of those Senators present. If the respondent is convicted on one or more of the articles against him or her, the Presiding Officer will pronounce the judgment of conviction and removal. No formal vote is required for removal, as it is a necessary effect of the conviction. The Senate has not always voted on every article of impeachment before it; for example, when the Senate did not convict President Andrew Johnson in the votes on three of the articles of impeachment against him, the Senate did not vote on the remaining articles.
The Senate may subsequently vote on whether the impeached official shall be disqualified from again holding an office of public trust. If this option is pursued, a simple majority vote is required.
From my reading, I interpret this to mean that the vote on disqualification from office is optionally considered after a successful conviction.
The source document cites III Hinds’ §2397; VI Cannon’s §512 in formulating this section on the judgement. This citation refers to precedence set in a prior impeachment case of Judge West H. Humphreys in 1862 where it was decided that these are two separate questions to be considered rather than taking it that conviction necessarily implies a disqualification from office.
Mr. Trumbull said:
I have very serious doubts whether it is a double question; whether the whole is not one judgment. ‘‘Judgment in cases of impeachment shall not extend further than to removal from office and disqualification to hold and enjoy any office of honor, trust, or profit under the United States.’’ I am not sure but that when the Constitution says it shall not extend further than that, it necessarily follows that it shall extend that far. It is not in the alternative, and I am by no means satisfied that that consequence does not necessarily follow the conviction. It is a limitation. As is well suggested by my friend from Pennsylvania [Mr. Wilmot], could you impose that latter part without the former? Could you decide that he should be disqualified to hold and enjoy any office of honor, trust, or profit? If each proposition is independent, it must be able to stand by itself without affecting any other. I am by no means satisfied that these are independent propositions. It seems to me that altogether the safer way is to take the question on them together.
Mr. Jacob Collamer, of Vermont, said:
Mr. President, I take it the test of the divisibility of a question depends upon whether there can be a vote left after it is divided, let the first be decided as it may. That is the criterion; that, if after you have voted ‘‘yea’’ or ‘‘nay’’ upon the first article of division, there is still a question to be decided if the decision be either way. Now, in this case, suppose the proposition to be that this man be deprived of office, and that he be rendered ineligible, and it is divided, and the vote shall be that he be not deprived of his office; is there anything left? There would be nothing left to vote on, because the rendering him ineligible hereafter is only a consequence of the first, and rests in judicial discretion
whether we put it on or not. It is not, in my apprehension, divisible, because a vote in one way on the first branch would render it impossible to get along with the second.
Mr. O. H. Browning, of Illinois, said:
We have the authority of an adjudicated case of the action of the Senate, in which they found a judge guilty upon impeachment and entered against him a judgment of ouster from his office; going no further. I apprehend it was competent for them to do that. They were not bound to attach to it the other consequence that may be attached to it under the Constitution, of disqualification forever
thereafter to hold office. It may frequently occur—it occurred in that case, it may occur again—that a majority of the Senators would feel it their duty to vote for his ouster from office, and would not feel it their duty to vote for his disqualification forever thereafter to hold any other office under the Government, however unimportant. If you are compelled to put the question, and the whole question, as one question—to put it all together—men who are unwilling to vote to disqualify him forever, disfranchise him forever, will be constrained to vote that he be ousted from office, and also to vote for another proposition, which in their judgments would be unjust. That would follow inevitably; and after you had taken the question on them jointly, I apprehend you could not return and divide them, and take the propositions separately, so as to say whether he should be ousted from office.
The President pro tempore said:
In the judgment of the Chair these are separate and divisible propositions. * * * From the authority of the Pickering case the Chair is obliged to say that it is a divisible proposition.
Elsewhere in this document, from a different impeachment, the following argument is made. This argument was made in regards to whether or not William W. Belknap could be impeached in 1876 as he had already resigned from his position. Your question is notable here because removal was a moot point—he had already resigned—but there was still a potential question for whether or not he could hold a public office in the future.
Mr. Jeremiah S. Black, also of counsel for the respondent, said:
... Removal is not only the object of impeachment, but it is the sole object. Removal and disqualification are so associated together that they can not be separated. You cannot pronounce a judgment of removal without disqualifying; and you can not pronounce a judgment of disqualification without removal, because the judgment which the Constitution requires you to pronounce is a judgment of removal and disqualification-not removal or disqualification; and this is made perfectly manifest to my mind from the experience we have had in Pennsylvania. It was thought by the convention that framed our Constitution desirable that the Senate, upon conviction of an offender of this kind, should have the discretion to say that he might be removed without being disqualified; and accordingly they
changed the provision which had previously been copied from the Constitution of the United States, and instead of saying what is said here, that judgment shall extend to removal and disqualification, it says it shall extend to removal, or to removal and disqualification. The effect of that was to allow of a judgment of removal alone, but not of disqualification alone—removal alone, or removal and
On the other hand, the managers for the House of Representatives maintained,
with careful citation of authorities, that impeachment was intended to reach a
public officer while in office or after he had left office. Mr. Manager Scott Lord
... It is supposed by many that because an officer must be removed no judgment can be pronounced without pronouncing the judgment of removal. This, it seems to me, is a very great error. If he is in office, of course under the Constitution he must be removed; but if out of office, the sentence of disqualification or some inferior sentence may be passed upon him, for the obvious reason that the sentence is divisible.
... I do not see, then, any possible view in which there is difficulty; and the learned counsel on the other side will not be able to create any difficulty excepting under the claim that a person in office, having so conducted himself as to be worthy of impeachment, finding that it is impossible to escape the facts or pervert them, may, I repeat, defeat the Constitution for the purpose of preventing his punishment.
Ultimately, in Belknap's trial, the Senate resolved that they had jurisdiction to try him for the impeachment, even though he had already resigned. When voting on the impeachment, the Senate did not reach the 2/3 threshold necessary for conviction on any articles, but they did attain a simple majority for each article. Thus, he was acquitted of all charges. It does not appear that they then went on to vote on whether or not he should be barred from future public office. This case is most important in showing that you can be impeached for actions taken while in office, even after you've left that office. However, beyond that, it looks like this case also supports the idea that the Senate will not vote on whether or not to bar the impeached individual from future public office if they fail to convict on any of the articles of impeachment.
I couldn't find anywhere that specifically states that votes on disqualification from future public office are contingent on a successful conviction on an article of impeachment, but it appears that arguments in prior cases in the Senate as well as their actions in these cases support this understanding.
To make a long answer even longer, we can look to the Senate Manual on the Rules for Impeachment Trials.
XXIII. An article of impeachment shall not be divisible for the purpose of voting thereon at any time during the trial. Once voting has commenced on an article of impeachment, voting shall be continued until voting has been completed on all articles of impeachment unless the Senate adjourns for a period not to exceed one day or adjourns sine die. On the final question whether the impeachment is sustained, the yeas and nays shall be taken on each article of impeachment separately; and if the impeachment shall not, upon any of the articles presented, be sustained by the votes of two-thirds of the Members present, a judgment of acquittal shall be entered; but if the person impeached shall be convicted upon any such article by the votes of two-thirds of the Members present, the Senate shall proceed to the consideration of such other matters as may be determined to be appropriate prior to pronouncing judgment. Upon pronouncing judgment, a certified copy of such judgment shall be deposited in the office of the Secretary of State. A motion to reconsider the vote by which any article of impeachment is sustained or rejected shall not be in order.
So regardless of any theoretical or precedential argument, the rules of the Senate indicate that the consideration of any further judgements in the impeachment trial (i.e. whether or not to disqualify the individual from holding public office) only commences if the individual is convicted.