But Mr. Trump’s staunchest defenders demanded that vote.

“For over eight months, they have had the opportunity to choose transparency. But they’ve instead chosen to withhold information and impede any effort of Congress to conduct oversight,” said Representative Mark Meadows of North Carolina, a sponsor of Thursday’s House resolution who raised the possibility of impeachment this week. “If Rod Rosenstein and the Department of Justice have nothing to hide, they certainly haven’t acted like it.”

Thursday’s hearing had been called to allow the law enforcement officials to field questions about the 500-page report from the department’s inspector general, Michael E. Horowitz. He found that the F.B.I. broke with longstanding policy in its handling of the Clinton email case, badly damaged its reputation and opened the door to accusations of political bias. The report criticizes at length two F.B.I. officials, Peter Strzok and Lisa Page, who exchanged voluminous texts disparaging Mr. Trump while they were helping to lead the investigation into both Mrs. Clinton and Mr. Trump’s campaign.

But in the hearing room, Republicans advanced a wide range of topics, including now-familiar and largely unsubstantiated charges against Mr. Rosenstein and the special counsel investigation that he oversees. One by one, the deputy attorney general sought to dispose of them.

Contrary to Mr. Trump’s repeated assertions — including in a Thursday morning tweet — Mr. Rosenstein said he was “not aware of any disqualifying conflict of interest” involving Mr. Mueller.

Representative John Ratcliffe, Republican of Texas, asked if Mr. Mueller had properly addressed the potential taint of political bias brought to the case by Mr. Strzok. Mr. Rosenstein said he was confident Mr. Mueller had taken “appropriate steps” to do so and would continue to.

Representative Jim Jordan, Republican of Ohio, pressed Mr. Rosenstein about news media reports saying that he had threatened members of the House Intelligence Committee staff with a subpoena during a heated dispute this year. Mr. Rosenstein disputed that, too, daring his accusers to contradict his denial under oath.