“The case for impeachment based on the Mueller investigation has been now publicly crystallized and articulated,” said Representative Jamie Raskin, Democrat of Maryland and a leading impeachment advocate on the Judiciary Committee. “The evidence is overwhelming: 10 different episodes of presidential obstruction of justice and the Trump campaign’s enthusiastic embrace of the Russian attack on our elections.”

The majority of the Democratic caucus remains skeptical about what it sees as a politically perilous push that would lead to an almost certain acquittal in the Senate and further drain attention from its legislative work. Some moderate Democrats who helped deliver the House majority to the party in 2018 by flipping Republican seats sounded increasingly exasperated.

“While there may be certain parts of it that are distasteful or disturbing, I don’t see the high crimes or misdemeanors,” said Representative Jeff Van Drew, who flipped a seat in southern New Jersey, discussing Mr. Mueller’s findings. “If we just let this overshadow all these other issues for a longer period of time, too long a period of time, we are really endangering the election for the Democrats.”

Democrats had never expected Mr. Mueller to cinch their case against the president, but they did hold out hope that testimony from the man who pursued Russian election interference and possible presidential obstruction of justice for 22 months would clarify their path forward. Those hopes faded by the time the House gaveled in on Thursday, and with lawmakers leaving Washington for a six-week recess, they now appear likely to fester into the fall.

Ms. Pelosi, an impeachment skeptic and a fierce guardian of members like Mr. Van Drew, did her best to keep the House investigations open. She touted the importance of Mr. Mueller’s testimony and urged lawmakers to “do your own thing” when it comes to impeachment. But behind closed doors, she took issue with those arguing that Democrats opposed to opening an impeachment inquiry are failing their responsibilities under the Constitution.