Key witnesses have renewed their definitive statements that they did not see any of the sexual misconduct that a few women have said they suffered at the hands of Supreme Court nominee Judge Brett M. Kavanaugh, bolstering his case as the FBI gets into the heart of its investigation.

President Trump made clear over the weekend that he remains firmly behind Judge Kavanaugh and urged voters to cast Republican ballots in key Senate races this year to protest the “angry and mean and nasty and untruthful” approach Democrats have taken toward the nomination.

Republicans on Sunday said absent any major new revelations, senators on the fence are likely to vote to confirm Judge Kavanaugh.

“If there’s some shocking new bit of, not accusation, but evidence and proof, then of course I’m open to evaluating that evidence,” Sen. Tom Cotton, Arkansas Republican, said on CBS’s “Face the Nation.” “But I strongly suspect that every statement that was made to the Judiciary Committee under penalty of perjury is the exact same statement that’s going to be made to the FBI.”

The FBI has until Friday to investigate claims by Christine Blasey Ford and other women who have come forward to claim they were victims of sexual misconduct that Judge Kavanaugh was involved in during his high school and college years.

With those allegations swirling, Sen. Jeff Flake of Arizona forced fellow Republicans to delay voting on the judge’s confirmation in order to give the FBI time to reopen the background investigation into Judge Kavanaugh.

Democrats praised Mr. Flake and excoriated Judge Kavanaugh, saying his indignant denials of allegations — which remain uncorroborated more than two weeks after they began to surface — rang hollow.

Sen. Mazie K. Hirono, Hawaii Democrat, said Judge Kavanaugh could have cleared up matters if he had demanded that the FBI reinvestigate himself.

“He could have helped us get over this whole issue of an FBI investigation,” Ms. Hirono said on ABC’s “This Week.” “He wouldn’t go there. But finally, finally we’re going to get one, and the question [is] are we going to get the kind of investigation that is thorough and fair?”

Despite the allegations, Judge Kavanaugh has cleared his first hurdle.

The Senate Judiciary Committee voted 11-10 along party lines Friday to advance the nomination to the full Senate floor.

But rather than a speedy vote there, action is stalled while the FBI conducts its investigation of Ms. Blasey Ford’s accusation that Judge Kavanaugh pinned her down and tried to force himself on her at a party more than 35 years ago when they were both in high school; Deborah Ramirez’s claim that he exposed himself to her during an alcohol-fueled party at Yale University; and Julie Swetnick’s less-developed claim that Judge Kavanaugh was complicit in gang rapes of girls in high school.

Judge Kavanaugh has strenuously and forcefully denied all the accusations and has said Ms. Swetnick’s allegation is beyond the pale.

An attorney for Ms. Ramirez confirmed she has been contacted by the FBI, but it’s unclear whether Ms. Swetnick herself will be questioned as part of the inquiry.

“How do you conduct a legitimate, fair & thorough investigation into allegations unless you interview the person actually making the allegations about her experiences, what she witnessed, and what facts and other witnesses she is aware of? Answer - YOU CAN’T. And that’s by design,” Michael Avenatti, an attorney for Ms. Swetnick and frequent thorn in the side of Mr. Trump, tweeted Sunday.

Ms. Blasey Ford has identified three people she says were at the party where she was assaulted, in addition to Judge Kavanaugh. None of those three has backed up her story.

Indeed, Leland Keyser, a longtime friend of Ms. Blasey Ford, told the Judiciary Committee over the weekend that she stands by her statement that she doesn’t remember any such party and doesn’t recall even having met Judge Kavanaugh.

She said she will cooperate fully with the FBI.

Mark Judge, the man Ms. Blasey Ford says was in the room when she was assaulted and who Ms. Swetnick says was part of the gang rapes, also said he will cooperate with the FBI. He repeated his denial of any knowledge of those crimes.

The New York Times reported that in addition to Ms. Ramirez and Ms. Keyser, the FBI will also likely question P.J. Smyth, another guest Ms. Blasey Ford says was at the 1982 party.

Mr. Trump over the weekend stood by his nominee, calling Democrats’ treatment of Judge Kavanaugh “angry and mean and nasty and untruthful.”

The president was stumping for Patrick Morrisey, West Virginia’s Republican attorney general, who is seeking to unseat Sen. Joe Manchin III, a Democrat.

Mr. Manchin is one of only two Democrats who have not yet signaled plans to vote against Judge Kavanaugh’s confirmation — but Mr. Trump said Mr. Manchin is enabling the other Democrats’ attacks on the nominee.

“I just hope you don’t sit home because bad things will happen if you sit home,” the president told a rabidly enthusiastic crowd.

Democrats on Sunday said Judge Kavanaugh should recuse himself from legal matters involving congressional Democrats if he is confirmed, citing his “partisan” performance during last week’s hearing.

“His animus toward the entire Democratic Party, toward people associated with it like the Clintons, should be disqualifying,” said Rep. Jerold Nadler, New York Democrat. “He showed no judicial temperament.”

Mr. Nadler, who is in line to become chairman of the Judiciary Committee if Democrats take control of the House, said the House would continue to investigate a Supreme Court Justice Kavanaugh if the Senate doesn’t do a thorough job.

“We would have to investigate any credible allegation, certainly of perjury and other things that haven’t been properly looked into before,” he said on “This Week.”

Republicans have said the performance from Judge Kavanaugh, who forcefully denied accusations of sexual assault and misconduct in high school and college, rang true from someone who believes he has been wrongly accused.

The judge said in his testimony last week that his opponents have engaged in a “calculated and orchestrated political hit, fueled with apparent pent-up anger about President Trump and the 2016 election.”

“Fear that has been unfairly stoked about my judicial record. Revenge on behalf of the Clintons. And millions of dollars in money from outside left-wing opposition groups,” he said.

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