Brett M. Kavanaugh made a last-minute appeal to senators Thursday to elevate him to the Supreme Court, and Republicans were increasingly hopeful of corralling the votes this weekend to do just that, after an FBI investigation failed to back up several women’s claims of sexual misconduct during his high school and college years.

Judge Kavanaugh wrote an op-ed for The Wall Street Journal defending his “too emotional” testimony to senators last week, saying he may have gone overboard but was desperate to clear his name of the sordid things he has been accused of.

“I said a few things I should not have said,” he admitted. “I hope everyone can understand that I was there as a son, husband and dad. I testified with five people foremost in my mind: my mom, my dad, my wife, and most of all my daughters.”

He vowed, though, to be an impartial justice, keeping “an open mind in every case.”

The unusual op-ed served as a capstone on a confirmation that has broken all the usual norms, setting a new low for judicial nominations and leaving both parties wondering where they go from here.

The immediate answer is to vote. Republicans hope to overcome a Democratic filibuster in a vote Friday morning. If they are successful, then a final confirmation vote will be held Saturday.

One wrinkle arose Thursday when Sen. Steve Daines, Montana Republican, told The Associated Press that he will be at his daughter’s wedding Saturday. If all other Republicans are present and vote for confirmation, then his absence won’t matter.

Eyes are on Republican Sens. Jeff Flake of Arizona, Susan M. Collins of Maine and Lisa Murkowski of Alaska, who have yet to publicly commit to backing the judge.

Neither has Sen. Joe Manchin III of West Virginia. He is the last remaining undecided Democrat, after Sen. Heidi Heitkamp announced Thursday that she will vote against Judge Kavanaugh. Analysts say the decision could seal her defeat in next month’s election in pro-Trump North Dakota.

Senators spent much of Thursday reviewing the FBI’s update to Judge Kavanaugh’s background check. Those 46 pages, from interviews with nine of the 10 people they reached out to, did not include any bombshell revelations.

Democrats complained that the review was incomplete, leaving dozens of important witnesses who were never interviewed. Republicans said the FBI made its own decisions — and two of the key Republicans, Mr. Flake and Ms. Collins, said they thought the FBI’s investigation was thorough.

Republicans said they felt good heading into Friday’s vote — though they were cautious about their predictions of success.

“I can just tell you how I feel about it — I think he will be confirmed,” said Sen. John Kennedy, Louisiana Republican.

“Hopefully, we’re 48 hours away from having someone new on the Supreme Court,” said Judiciary Committee Chairman Charles E. Grassley of Iowa.

Protesters filled the halls of Senate office buildings, hoping their anti-Kavanaugh chants would reverberate with the undecided senators, or alternatively serve as a political warning to the 48 Republicans believed to be committed to voting for Judge Kavanaugh.

Capitol Police arrested at least 302 of the protesters.

One protester, who was a sexual assault survivor, aggressively questioned Mr. Manchin as he waited for an elevator about how he will vote.

“How long is it going to take you to listen to us?” the protester asked.

“I am listening to you,” Mr. Manchin responded.

But the West Virginia Democrat said he hasn’t finished reviewing the FBI report and plans to return to do so Friday morning ahead of the vote, which is scheduled for 10:30 a.m.

Ms. Collins spent hours in the secure room where the FBI report was held. She emerged around 6:30 p.m. to tell reporters that she had finished reading but did not hint at how she plans to vote.

Republican leaders had hoped to have confirmed Judge Kavanaugh a week ago, but the process was delayed after Mr. Flake demanded that the FBI step in to try to sort out the he-said, she-said nature of the assault allegations from decades ago.

Sen. Jeff Merkley, Oregon Democrat, is trying to force yet another delay.

He asked a federal judge Thursday to issue a ruling ordering the government to produce documents from Judge Kavanaugh’s past, hoping that such an order would force the Senate to postpone a final Kavanaugh vote.

He has accused President Trump of breaking the Constitution by not turning over the information fast enough.

The Justice Department countered that Mr. Merkley already has a way to register his disapproval with the process: He can join the Democratic filibuster and, should that fail, exercise his constitutional right as a senator and vote against the nomination.

Mr. Trump, who was out campaigning for Republican candidates, said the battle over Judge Kavanaugh has invigorated the party with a month to go before the midterm congressional elections.

“Look at the polls over the last three or four days,” the president said at a rally in Minnesota. “It shows [Democrats’] rage-fueled resistance is starting to backfire at a level nobody’s ever seen before. We love it.”

Polling suggests pro-Kavanaugh sentiment has helped close the Republicans’ enthusiasm gap with Democratic voters, and several Republican Senate campaigns said they are seeing big boosts in their races to unseat Democratic incumbents.

That may be particularly true in North Dakota, where Ms. Heitkamp now trails by double digits in her re-election bid against Rep. Kevin Cramer, her Republican challenger. Voters tell pollsters they are prepared to punish Ms. Heitkamp if she votes against Judge Kavanaugh.

She told WDAY-TV that she wouldn’t be swayed by her political future.

“If it was a political decision for me, I would certainly be deciding this the other way,” she said. “I can’t get up in the morning and look at the life experience I’ve had and say ‘yes’ to Judge Kavanaugh.”

• Dave Boyer, S.A. Miller and Gabriella Muñoz contributed to this article.

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