Even some of Mr. Biden’s friends say he will have to show remorse for the Hill-Thomas hearings if he wants Democrats to accept him as a leader for the future in a party sharply defined by the concerns of liberal women.

Former Senator Barbara Boxer of California — who won her seat in the so-called “Year of the Woman” election in 1992, spurred in part by outrage over the Hill-Thomas hearings — said Mr. Biden had to address his past head-on.

“If he handles it right, it could be a plus; if he handles it wrong, it’ll be a minus,” Ms. Boxer said. “And handling it right means stepping up to the plate.”

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Mr. Biden’s low profile in the Kavanaugh fight has contrasted sharply with the prominence of other potential 2020 candidates, including Senators Kamala Harris of California and Cory Booker of New Jersey, who questioned the judge and Dr. Blasey in the Judiciary Committee, and Elizabeth Warren of Massachusetts. Ms. Warren said Saturday that the Republicans’ treatment of Dr. Blasey had pushed her closer to running for president.

Mr. Biden has privately expressed frustration about the ire directed at him for his Judiciary Committee tenure, according to multiple people who have spoken with him directly but requested anonymity to discuss private conversations. He has been incensed about Republicans’ taking some of his past remarks out of context: They brandished one video clip from 1991 to suggest Mr. Biden had dismissed the value of having the F.B.I. investigate Ms. Hill’s allegations. Mr. Biden’s spokesman said he had been rejecting an argument from a Republican, Senator Orrin Hatch of Utah, who said the F.B.I. had effectively exonerated Mr. Thomas.

But Mr. Biden has also complained to allies that fellow Democrats are now judging his role in the Hill hearings without, he believes, appropriately recognizing the limits of his power as a committee chairman, and the political reality of that moment.