The presidential payback machine kicked into high gear Thursday as President Trump and his Republican allies sought to hold accountable opponents in government and the media who propelled phony allegations of collusion with Russia.

“We’ve got to hold the people to account who did it, as a deterrent to anybody doing it again,” Trump attorney Rudolph W. Giuliani told The Washington Times after meeting with the president. “This wasn’t an accident. This was a deliberately conceived plot that was carried along in a malicious way.”

Mr. Trump and congressional Republicans demanded the ouster of Rep. Adam B. Schiff, California Democrat and chairman of the House Permanent Select Committee on Intelligence. Mr. Schiff, an unrepentant promoter of the Russia accusations despite special counsel Robert Mueller’s conclusion that no Americans conspired with Moscow’s efforts to meddle in the 2016 election.

Mr. Schiff “should be forced to resign from Congress,” said a Twitter message from Mr. Trump. He said the lawmaker “spent two years knowingly and unlawfully lying and leaking.”

The president and his supporters also are targeting former CIA Director John O. Brennan, former FBI Director James B. Comey, former FBI Deputy Director Andrew McCabe, former FBI agent Peter Strzok, House Judiciary Committee Chairman Jerrold Nadler, former Attorney General Loretta E. Lynch and others.

They also are calling into question whether President Barack Obama knew about or approved the FBI investigation of the Trump campaign, which began in summer 2016. Mr. Obama was campaigning hard for Democrat Hillary Clinton to beat Mr. Trump.

Mr. Giuliani said in an interview that he believes the Justice Department is capable of investigating its own actions, possibly with a task force of hand-picked agents “who don’t have significant political involvement.”

“It should be done by the Justice Department and the FBI, because they should be given a chance to straighten themselves out,” he said. “I know that the vast majority of the Justice Department, whether they’re Republicans, Democrats or independents, are honest people. These are just the politicians at the top that were corrupted.”

Mr. Giuliani said he hasn’t seen the confidential Mueller report. The president told reporters that he hasn’t seen the report either, although he said it has a “beautiful conclusion.”

“I wish you could have got it in one week instead of taking two years, but the result was great: no obstruction, no collusion, no anything,” Mr. Trump said on his way to a campaign rally in Michigan.

The Mueller report’s finding of no collusion “really makes the vindication that much more powerful,” Mr. Giuliani said, because Mr. Mueller’s team included “eight or nine people who are highly partisan Democrats.”

“These people tried every dirty trick you could to try to catch” Mr. Trump, Mr. Giuliani said. “And they couldn’t, which means it didn’t happen.”

He said an investigation of the origins of the FBI’s Russia probe is “very, very necessary, because as the president says, we have to make sure this doesn’t happen again.”

“That’s a very, very serious thing: to create, support, maintain and exaggerate false charges against any president of the United States,” Mr. Giuliani said. “You’re hurting the country. The people who were involved in it and did things like sign false affidavits, omit facts that they normally wouldn’t or change facts, the most generous thing I could say about them is they lost all perspective. They may be more educated, but they were affected by the same psychological syndrome that drove people, after President Trump was elected, to go to psychiatrists.”

In an appearance on Fox News with Sean Hannity on Wednesday night, Mr. Trump questioned aloud whether Mr. Obama knew of or approved the investigation of his campaign.

“Why didn’t President Obama and the Obama administration do something when they were informed about Russia and the election?” the president said.

A spokesman for Mr. Obama didn’t respond to a request for comment. Mr. Obama has said he told Russian President Vladimir Putin to “cut it out” during a face-to-face meeting in September 2016.

Mr. Trump said he will leave it to Attorney General William P. Barr and Senate Judiciary Committee Chairman Lindsey Graham, South Carolina Republican, to decide on any investigations into the origins of the government’s probe of the Trump campaign. Mr. Graham has called for a second special counsel to look into the matter and said he plans to call witnesses such as Mr. Comey to answer questions in Judiciary Committee hearings.

Sen. Rand Paul, Kentucky Republican, also has called for an investigation of former Obama administration officials “who concocted and spread the Russian conspiracy hoax.”

Steve Bannon, a former White House strategist, has suggested the creation of a bipartisan select committee, akin to the Church Committee in 1975 that investigated abuses by the CIA, FBI, National Security Agency and IRS. He has predicted that Mr. Trump will go “full animal” on those who pushed the Russia accusations.

But some political strategists are pointing to polls to warn that Mr. Trump could hurt his re-election chances next year if he appears to be seeking retribution for the Russia probe. A Morning Consult/Politico poll released immediately after the disclosure of the Mueller report’s conclusions showed that independent voters were 9 percentage points less likely to vote for Mr. Trump, despite his vindication on the collusion allegations.

A Reuters/Ipsos survey released Wednesday found that 48 percent of respondents said they believe that Mr. Trump “or someone from his campaign worked with Russia to influence the 2016 election.” That number was down from 54 percent a week before the Mueller report’s conclusions were released.

Republican strategist Karl Rove said rather than focusing on retribution, Mr. Trump and his team “should use the Mueller report to pivot to issues, like the economy and the opioid crisis, that matter to swing voters who will decide the 2020 presidential election.”

“The party that mounts the better effort to persuade swing voters will likely be the winner of the coming presidential election,” he wrote in The Wall Street Journal.

Mr. Giuliani said he isn’t worried about political blowback against the president from an investigation targeting the people who launched the Russia probe.

“No,” he said. “This has to be done, and it’s so important to the country that it be done, that I think you can explain that to people. It should never be done to another president.”

He said of Mr. Trump, “I’ve known him for 30 years. He’s capable of handling this. It was hard on him, but he was capable of handling it. I’ve been with him on very bad days and very good days, and he can function no matter what they throw at him.”

Justice Department Inspector General Michael E. Horowitz already is looking into potential surveillance abuses by the FBI, a review that began in March 2018. He criticized senior FBI officials in a previous report for a “willingness to take official action” to hurt Mr. Trump’s chances of becoming president.

U.S. Attorney for Utah John Huber was appointed a year ago by the U.S. attorney general to review surveillance abuses by the FBI and Justice Department, as well as authorities’ handling of a probe into the Clinton Foundation.

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