Deputy Attorney General Rod Rosenstein refused to say Thursday whether he read the secret surveillance warrant used to snoop on a Trump campaign figure last year, frustrating congressional Republicans who urged him to push for an end to the Russia investigation.

Mr. Rosenstein sat for a five-hour grilling before the House Judiciary Committee, where he fended off questions about the role that anti-Trump FBI agent Peter Strzok played in starting the Russia investigation, how long Robert Mueller’s investigation would last and whether he would hand over sensitive documents that Congress requested.

The hearing, which also included FBI Director Christopher A. Wray as a witness, comes as tensions rise between House Republicans and the Justice Department.

At one point, lawmakers had to take a break to go the House floor to vote on a resolution setting a July 6 deadline for the documents to be turned over. The deadline doesn’t include a punishment, though some lawmakers have said contempt of Congress or even impeachment could follow if the FBI and Justice Department won’t meet House committees’ demands.

“Why won’t you give us what we’ve asked for?” Rep. Jim Jordan, Ohio Republican, demanded of Mr. Rosenstein.

He said Congress has “caught you hiding information.”

Mr. Rosenstein fired back by saying the dispute over documents was a matter of FBI policies and secrecy, not an attempt to thwart Congress.

“Your use of this to attack me personally is deeply wrong,” he said. “When you find some problem with the production with questions, it doesn’t mean that I’m personally trying to conceal something from you.”

Mr. Jordan then asked Mr. Rosenstein to explain a Fox News report this month that said he had threatened to subpoena cellphones, emails and other documents from staffers of the House Permanent Select Committee on Intelligence.

The Deputy Attorney General said the media reports were “mistaken.”

Rep. Trey Gowdy, South Carolina Republican, urged Mr. Rosenstein to push Mr. Mueller to wrap up the Russia investigation, saying it hasn’t uncovered evidence that President Trump did anything wrong.

“We’ve seen the bias; we need to see the evidence,” Mr. Gowdy said. “If you have evidence of wrongdoing by any member of the Trump campaign, present it to the damn grand jury. If you have evidence that this president acted inappropriately, present it to the American people.

“Whatever you got, finish it the hell up, because this country is being torn apart,” he said.

Mr. Rosenstein cautioned against drawing any conclusions from what is little-known publicly about the special counsel’s investigation, saying “the best thing we can do is finish it appropriately.”

He also faced scrutiny over his own role in assisting the FBI in its investigation into Trump campaign figures.

Mr. Rosenstein last year approved the extension of a Foreign Intelligence Surveillance Act warrant on Carter Page, who had been a foreign policy adviser to the campaign.

On Thursday, Mr. Rosenstein refused to say if he had read the warrant when he signed it. He would say only that if the Justice Department inspector general found he did something wrong would he accept that conclusion.

Rep. Ron DeSantis, Florida Republican, suggested that Mr. Rosenstein should recuse himself from the Russia investigation. He jokingly referred to it as the “Rosenstein investigation” and noted that the deputy attorney general himself appointed and is supervising Mr. Mueller.

“But you wrote the memo saying that Comey should be fired, and you signed the FISA extension for Carter Page,” Mr. DeSantis said. “So my question to you, it seems like you should be recused from this more so than [Attorney General] Jeff Sessions just because you were involved in making decisions affecting both prongs of this investigation.”

“Congressman, I can assure you that if it were appropriate for me to recuse, I would be more than happy to do so,” Mr. Rosenstein joked. “But it’s my responsibility to do it.”

He and Mr. Wray defended their handling of Congress‘ document requests.

The FBI director told the committee that 880,000 documents have been produced and 100 staffers were working around the clock, including on weekends, to respond the request.

Congress voted along party lines, 226-183, to impose the July 6 deadline for the documents.

“When we get these documents, we believe it will do away with this whole fiasco of what they call the Russian-Trump collusion because there wasn’t any,” Rep. Mark Meadows, the North Carolina Republican who sponsored the deadline resolution, said on the House floor.

Democrats called the document deadline and the hearing distractions.

They pressed Republicans to release the transcript of an all-day deposition this week with FBI agent Peter Strzok, who was removed from the Mueller investigation after the inspector general found he had sent vicious anti-Trump texts to his paramour, an FBI lawyer.

“The American people deserve to hear Peter Strzok’s testimony under oath,” said Rep. Ted Lieu, California Democrat. “Do not hide his testimony.”

Judiciary Committee Chairman Bob Goodlatte, Virginia Republican, said the testimony will be released to the American people at the appropriate time.