The new filing comes two days after Mr. Mueller testified before Congress for the first time about the findings of his 22-month investigation into Russian interference in the 2016 election and possible obstruction of justice by Mr. Trump. Republicans — and some Democrats — said Mr. Mueller’s lackluster appearance had all but ended the impeachment threat, and they were not convinced the committee’s actions on Friday had changed that.

“Democrats want to convince their base they’re still wedded to impeachment even after this week’s hearing, but a baseless legal claim is an odd way to show that,” said Representative Doug Collins of Georgia, the ranking Republican on the Judiciary Committee. He predicted that the legal maneuver would fail.

But Democrats who control the panel called Mr. Mueller’s testimony an “inflection point” and are now seeking to add more evidence about what they believe to be serious wrongdoing by Mr. Trump.

Mr. Nadler said the committee would continue the investigation during the House’s six-week summer recess, calling additional witnesses and filing a lawsuit as soon as next week to force Donald F. McGahn II, the former White House counsel, to testify unless he agrees to come voluntarily. Mr. McGahn’s account of presidential behavior sits at the center of Mr. Mueller’s report.

The 448-page document showed that the Trump campaign welcomed illegal assistance from the Russians in 2016 and expected to benefit from it, but investigators did not establish that Mr. Trump had conspired with them in the illegal hacking and dumping of Democratic emails.

It also explored several episodes in which Mr. Trump tried to impede the investigation. But the special counsel decided not to render judgment about whether Mr. Trump should be charged with obstruction of justice, citing a Justice Department view that sitting presidents are temporarily immune from indictment while they are in office.

Aside from the coming court case, it is unclear whether the Judiciary Committee’s assertion that an impeachment investigation is effectively already underway will resolve a fight that has divided the Democratic Party. Around 100 House Democrats have said they support opening impeachment proceedings — including several more Friday morning — leaving about 135 who have not come out in support of such an inquiry or who outright oppose it.