Meanwhile, impatience is growing among the party’s liberal wing to move forward on impeaching Mr. Trump, as activists vent their frustration about a House Democratic majority they charge is too timid to effectively challenge the president.

Leaders have privately counseled Democratic lawmakers to use their August break to play up their policy agenda, holding events that highlight their accomplishments in the first 200 days. Those include a bid to raise the federal minimum wage to $15 an hour and measures to protect health care access. In a news conference on Thursday on the steps of the Capitol, Ms. Pelosi said Democrats would “own August” and ratchet up pressure on Mr. McConnell to take up legislation that the House has passed. The word “impeachment” was not uttered.

But progressive groups intend to amplify their pro-impeachment message and to intensify pressure on Democrats to fall in line. The liberal organizing group Stand Up America is planning a major mobilization next month to put Democrats on the spot on the subject, encouraging voters across the country to demand that their lawmakers support opening an inquiry.

Activists are particularly focused on the 30 or so progressive lawmakers who have not yet come out in favor of impeachment, as well as holdouts on the House Judiciary Committee, more than half of whom already back the move.

“During the August recess, we will ensure that every member of Congress hears from their constituents on why it’s the only path forward,” said the group’s president and founder, Sean Eldridge. “Right now, Congress is the only institution that can hold Trump accountable for his crimes — and it’s time for Speaker Pelosi to do the right thing by starting an impeachment inquiry.”

The satirical website The Onion on Thursday reflected the degree of liberal outrage at the speaker’s approach, blaring the headline, “Pelosi Concerned Outspoken Progressive Flank of Party Could Harm Democrats’ Reputation as Ineffectual Cowards.” It appeared just hours before Ms. Pelosi scored a major coup, winning passage of the breakthrough two-year budget agreement to raise the statutory limit on national debt and increase spending limits, with minimal defections by progressive Democrats.

The spoof pointed to very real frustration among activists, who argue that despite Democrats’ best efforts to brand themselves as pragmatic and in touch with kitchen-table concerns, they instead come across as feckless and ineffective by failing to establish a clear position on how to take on Mr. Trump.