Asked by Representative Bob Goodlatte, the Virginia Republican who chairs the committee, why he remained satisfied with Mr. Mueller, Mr. Rosenstein replied:

“Based upon what I know, I believe Director Mueller is appropriately remaining in his scope and conducting himself appropriately, and in the event there is any credible allegation of misconduct by anybody on his staff, that he is taking appropriate action.”

Mr. Rosenstein’s stance signaled that despite the mounting assault on Mr. Mueller by Mr. Trump’s supporters, the fundamental dynamic surrounding the special counsel had not changed: If Mr. Trump were to try to fire Mr. Mueller based on any developments so far, the president would likely first have to fire or force the resignation of Mr. Rosenstein and then hunt for a replacement willing to carry out his orders, echoing Richard Nixon’s so-called Saturday Night Massacre during the Watergate scandal.

Republicans repeatedly pressed Mr. Rosenstein to appoint a second special counsel to investigate political partisanship in the department in its handling of the Trump-Russia investigation or in last year’s decision not to charge Mrs. Clinton with a crime over her use of a private email server while secretary of state — an idea that has been promoted heavily by commentators on Fox News and elsewhere in recent days.

Mr. Rosenstein said he could not appoint another special counsel without a credible allegation of a potential crime to investigate.

The campaign against the special counsel, at the very least, provides a rallying cry for the president’s supporters to counter the drumbeat of news about Russian interference in the election and possible links to the Trump campaign. And a move by the Justice Department to show reporters the text messages that are the subject of an ongoing inspector general’s inquiry served to fuel the Republican campaign against Mr. Mueller.

Mr. Rosenstein confirmed that in addition to sending the messages to Congress the night before his testimony, the Justice Department invited reporters to view the messages it had given to lawmakers. That was a rare step, although officials in previous administrations have sometimes done so to avoid selective or misleading leaks from Capitol Hill.