Why has Pelosi not held the impeachment vote the White House is demanding?

Question

It's established that the House does not need to vote in order to being an impeachment inquiry, because a previous House had changed the rules so the committee already had subpoena power. However, the White House has said repeatedly (and eventually sent a letter) saying they wouldn't cooperate without such a vote.

What reasons has Pelosi (or other important Democrats) given for not holding a vote, as a way to negate the White House's ability to use this as a defense? Presumably it would be a party-line vote, which the Democrats would win, and then it would be a "properly authorized" inquiry.

Answer 1

Update: It seems that Pelosi has decided to go ahead with a resolution after all. The resolution will be introduced on Tuesday 10/29/19 and will be voted on by the full House on Thursday 10/31/19:

“This resolution establishes the procedure for hearings that are open to the American people, authorizes the disclosure of deposition transcripts, outlines procedures to transfer evidence to the Judiciary Committee as it considers potential articles of impeachment, and sets forth due process rights for the president and his counsel,” Ms. Pelosi said in a statement.

“We are taking this step to eliminate any doubt as to whether the Trump administration may withhold documents, prevent witness testimony, disregard duly authorized subpoenas, or continue obstructing the House of Representatives,” Ms. Pelosi wrote.

Shifting Course, Democrats Plan First Floor Vote on Impeachment Inquiry

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Since we can’t read Pelosi’s mind, any answer will be somewhat speculative, but here are 2 likely reasons why Pelosi is unlikely to do this.

It shows that Trump can push the House around

As has been covered in other questions, there is no need for a vote to start an impeachment investigation. This is just an excuse for the White House to ignore requests they don’t want to comply with, and if Pelosi agreed to this condition, then there’s no reason they wouldn’t just add a new one. By agreeing to Trump’s demands, she legitimizes his position that the White House can influence and control the investigation and is likely to lead to further attempts to do so. It’s a much stronger position to maintain that the Constitution gives impeachment power wholly to the House and the process will be carried out in accordance with House rules.

The California Democrat’s decision comes with some risks, such as continuing to give Republicans something to hang their qualms about the probe on, and the prospect that President Trump would follow through on the White House’s ultimatum to refuse to cooperate with the inquiry until there is a vote. But Pelosi believes he’s bluffing anyway.

“We’re not here to call bluffs. We’re here to find the truth, to uphold the Constitution of the United States,” Pelosi told reporters Tuesday night.

...

"If we do vote, Republicans will just come up with another process argument,” the Democratic aide with knowledge of the conversations told The Fix. “So calling their bluff is pointless. Trump will not cooperate no matter what.”

Why Nancy Pelosi doesn’t feel much pressure to hold a vote on the impeachment inquiry - The Washington Post

With this in mind, the recent ruling by Judge Beryl Howell that the impeachment inquiry does not require a floor vote is likely an important contributor to Rep. Pelosi's decision to hold a preliminary impeachment vote after all:

Democrats have countered that no resolution is required under the Constitution or House rules and pointed out that impeachment efforts to remove other officials, like judges, started without such a vote.

Judge Howell agreed with the Democrats, calling the Republican arguments “cherry-picked and incomplete” and lacking support from the Constitution, House rules or court precedents.

“Even in cases of presidential impeachment, a House resolution has never, in fact, been required to begin an impeachment inquiry,” wrote Judge Howell

Impeachment Inquiry Is Legal, Judge Rules, Giving Democrats a Victory

Now that a judge has affirmed that the House does not need to pass a resolution authorizing an impeachment inquiry, Pelosi can choose to hold one without it looking like she was forced to by the Trump Administration.

A party line vote isn’t ideal

As you say, if they had a vote right now, it would almost certainly pass on a party line vote. That’s not ideal for 2 reasons. First, it makes it easier to paint this as a wholly partisan process, and thus makes the investigation easier to ignore. Secondly, it forces Republicans to take a firm stand on this issue, which would make it harder for them to change their minds later as more evidence comes out. For evidence of this, the best I could find from Pelosi herself is this quote where she asserts that coming out firmly for or against impeachment carries risks for Republicans which many would like to avoid:

Pelosi said on Friday that she believes the politics are most problematic for the minority, because a vote would put Republicans on the record about what's being revealed about Trump's requests for foreign governments to get involved in U.S. politics.

"You know who is most scared of having a vote on the floor – the Republicans," Pelosi said in Atlanta.

Trump Challenges House On Impeachment Vote. Pelosi Says She's Unmoved - NPR

Obviously, Pelosi isn’t really worried about protecting vulnerable Republican representatives. But, I think it supports the view that she believes that (at least some) Republicans are not 100% comfortable about siding with Trump (at least currently).

If there are Republicans that are wavering now, then future events may cause greater shifts. The hope (which may or may not occur) is that as the case against the president becomes stronger, and public opinion turns, some Republicans will put country (or at least their re-election prospects) ahead of party and vote to impeach (as some Democrats did with Clinton), thus giving the process more bipartisan legitimacy.

BBC reporter Anthony Zurcher also supports this view:

Ms Pelosi could also be hoping that the longer the investigation grinds on, the greater the chance Democrats could uncover that damning bit of evidence that breaks Republicans ranks. She may believe that it would be easier for Republicans to support impeachment if they weren't on the record voting against an investigation.

Why won't Democrats vote to authorise impeachment? - BBC

Answer 2

I think divibisan covered most of it. His points are reflected in a few more quotes from Democratic representatives quoted by a Politico article; alas none are from Pelosi herself. However, that article also raises an interesting point/reminder:

In the past, when judges have suggested the House should take a vote, Pelosi has responded by doing just that, taking action to preempt unfavorable rulings in court.

Earlier this year, Neomi Rao, a Trump-appointed federal appeals court judge, suggested that a full House vote would be necessary in order to buttress the House Oversight Committee’s subpoena seeking Trump’s financial records. Almost immediately, the House passed a largely symbolic resolution that puts the weight of the full House behind their subpoenas.

Apparently, in the most recent court cases, Pelosi felt no need to do that again. Judge Howell ruled in favor of the HoR (and so against the WH) in the Mueller grand-jury documents case without any other House resolution, simply based on the recommendations found in the Mueller report itself. Also,

In a double victory for Speaker Nancy Pelosi, Judge Beryl Howell — the chief federal judge in Washington — ruled that the impeachment inquiry Democrats have launched is valid even though the House hasn't taken a formal vote on it. The decision rejects arguments by DOJ and congressional Republicans that a formal vote is necessary to launch impeachment proceedings.

"Even in cases of presidential impeachment, a House resolution has never, in fact, been required to begin an impeachment inquiry," Howell determined, dismissing GOP arguments as unsupported.

As a side note,

She adds, "Tipping the scale even further toward disclosure is the public's interest in a diligent and thorough investigation into, and in a final determination about, potentially impeachable conduct by the President described in the Mueller Report."

Answer 3

As divibisan pointed out, Pelosi has just announced that the House will hold a vote after all. In her "Dear Colleague" letter (full text), she calls out her official reasons for not holding a vote before, which I've quoted below. It appears to boil down to "we didn't need to, so we didn't". (I've also reformatted to break it up into separate points.)

Of course, [the White House's] argument has no merit. The Constitution provides that the House of Representatives "shall have the sole Power of Impeachment." Multiple past impeachments have gone forward without any authorizing resolutions.

Just last week, a federal court confirmed that the House is not required to hold a vote and that imposing such a requirement would be "an impermissible intrusion on the House's constitutional authority."

More than 300 legal scholars have also refuted this argument, concluding that "the Constitution does not mandate the process for impeachment and there is no constitutional requirement that the House of Representatives authorize an impeachment inquiry before one begins."