But they have come under intense criticism from Republicans for failing to seek formal authorization for the inquiry, a step that is not required by the Constitution or House rules. In scheduling a vote now, Democrats were effectively challenging Mr. Trump and his congressional allies, who have called the inquiry the inquiry unfair and the process a sham, but avoided any substantive discussion of the president’s conduct.

Still, Republicans signaled that after weeks of calling for a vote on the inquiry, they would oppose the resolution en masse.

“We will not legitimize the Schiff/Pelosi sham impeachment,” Representative Kevin McCarthy, Republican of California and the minority leader, said in a tweet.

Stephanie Grisham, the White House press secretary, said Ms. Pelosi was “finally admitting what the rest of America already knew — that Democrats were conducting an unauthorized impeachment proceeding, refusing to give the president due process, and their secret, shady, closed door depositions are completely and irreversibly illegitimate.”

Democrats said their inquiry has been proper from the start. Ms. Pelosi reiterated what Democrats have argued for weeks and a Federal District Court judge ruled last week: that they did not need a formal vote of the full House to start a legitimate inquiry. (The Justice Department separately announced Monday it would appeal the ruling handed down Friday.)

So far, the work of the impeachment inquiry has mostly been done out of public view, with staff for Democrats and Republicans questioning a growing roster of diplomats and other administration officials in the closed chambers of the House Intelligence Committee. Democrats are pleased with the portrait they have assembled of a president who bypassed the normal channels of diplomacy to pressure Ukraine to investigate former Vice President Joseph R. Biden Jr. and unproven theories that could exonerate Russia from aiding his campaign in 2016 and implicate Democrats in interfering in the election instead.

That work was briefly interrupted on Monday, when Charles M. Kupperman, the former deputy national security adviser, defied a House subpoena for testimony, angering Mr. Schiff. The White House said on Friday that Mr. Kupperman, as one of the president’s “closest confidential” advisers, was immune from testifying, and directed him not to appear in defiance of a subpoena. That prompted him to file a lawsuit against Mr. Trump and congressional Democrats asking a federal judge whether he could testify, raising the prospect of a drawn-out legal battle over weighty questions about the separation of powers that could effectively stall the impeachment inquiry for months.