The release of a declassified memo on FBI surveillance abuses against the Trump campaign prompted calls Friday for Deputy Attorney General Rod Rosenstein to explain his involvement to Congress or resign.

The memo from Republicans on the House Intelligence Committee shows that at least one questionable surveillance warrant application was signed by Mr. Rosenstein, who already had a difficult relationship with President Trump. The memo said information supporting the application was obtained from a partisan anti-Trump dossier funded by Hillary Clinton’s campaign and the Democratic Party.

Rep. Ron DeSantis, Florida Republican, said Mr. Rosenstein will likely have to appear before Congress to explain his actions regarding the Foreign Intelligence Surveillance Act (FISA) warrants for members of the Trump campaign.

“I think Rosenstein is going to have to come to the Congress and explain his role in extending it,” Mr. DeSantis said on Fox News. “I mean, did he go back and review it and was satisfied, or he just extended? And is he going to be able to justify this as a proper use of FISA?”

Asked Friday whether he has confidence in Mr. Rosenstein, the president said dismissively, “You figure that one out.”

Attorney General Jeff Sessions defended his second-ranking official, saying Mr. Rosenstein represents “the kind of quality and leadership that we want in the department.”

The conservative Tea Party Patriots Action said it will pay for a television ad airing Monday that calls on Mr. Rosenstein to “do his job or resign.”

The ad calls him “a weak careerist at the Justice Department — protecting liberal Obama holdovers and the deep state, instead of following the rule of law,” according to a transcript released by the group.

But experts say that Mr. Rosenstein will likely weather the storm.

James Wedwick, who now runs an investigations firm, applied for surveillance warrants during his 35 years with the FBI.

He said the FBI was “irresponsible” in presenting evidence to the FISA court because it did not disclose the dossier’s origins. Anytime he asked the court for a surveillance warrant, he was required to present the court with the absolute best and the absolute worst of his information, including detailing concerns he had his information.

“I’ve put Mickey Mouse stuff in the affidavit because I was worried about leaving anything out,” Mr. Wedwick said. “So when I hear that they left out the whole tale about how they knew where the dossier came from, that judge should feel like the court has been defrauded.”

He did not blame Mr. Rosenstein for the court being presented with incomplete evidence, however.

“This started before he got there,” Mr. Wedwick said. “This thing got started under [former FBI director James B.] Comey’s and [former Deputy Attorney General Sally] Yates’ watch. The system was already up and running and other people had already signed off on this, so Rosenstein may not have given this the scrutiny it deserved. He has a good reputation. so if I were to give anyone the benefit of the doubt, it would be him.”

Richard Painter, the chief ethics lawyer for President George W. Bush, said the public outcry if Mr. Rosenstein is fired could prevent the president from pulling the trigger.

“If he wants to fire Rosenstein, it will get ugly,” Mr. Painter said, referring to Mr. Trump. “He’ll just dig himself into a bigger hole with obstruction accusations because it will seem like he’s firing Rosenstein to get at [special counsel Robert] Mueller. The advice I would give him is to leave it alone. If the president just stops, that could minimize his exposure to possible obstruction of justice.”

Mr. Painter said it was odd that Congress has not yet spoken with the judge who approved the FISA warrant.

“I would think the judge would have been subpoenaed by now,” he said. “You really can’t reach conclusions about the memo until we hear from the judge. If this is half-baked, where is the judge screaming that his court was mislead?”

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