On Thursday, more groups affiliated with the religious right piled on: Concerned Women of America’s legislative action committee sent a blast to its members urging “No More Delays,” and the American Family Association sent out another, “Confirm Kavanaugh Now!”

The importance of the Supreme Court to the Trump White House and the Republican Party is difficult to overstate. Mr. Trump has heralded Justice Neil M. Gorsuch and Judge Kavanaugh, his two Supreme Court nominees, as crowning achievements in an otherwise uneven presidency. Conservative groups have spent tens of millions of dollars building the men up as legal luminaries, gentleman scholars and the fulfillment of Mr. Trump’s campaign promise to nominate judges who have “a record of applying the Constitution just as it was written,” as one ad by the Judicial Crisis Network described Judge Kavanaugh.

A relatively smooth, predictable confirmation fight has also been a key part of Republicans’ strategy to keep the Senate. In the 10 states that Mr. Trump won where Democratic senators are up for re-election, Republicans have attacked Democrats for either opposing the judge or remaining noncommittal. But Dr. Blasey’s claims may have given Democrats who were on the fence a way to vote no without paying a steep political price.

Even social conservatives who describe Dr. Blasey’s account as part of a Democratic plot to upend the nomination acknowledge the bind they are in. While they decry the process as tainted and unfair, some are also arguing that they cannot be indifferent and insensitive to a victim.

“The worst thing that can ever happen to any woman or man who has been a victim is to shut them down and not listen to them,” said Marjorie Dannenfelser, president of the Susan B. Anthony List, a group that opposes abortion rights. “A tragic piece of this is people who will use that pain for an agenda. That is so clearly what is happening now.”

In the days since Dr. Blasey went public in an interview with The Washington Post and alleged that, when they were both teenagers, Judge Kavanaugh pinned her down on a bed, clapped his hand over her mouth and groped her, Republican leaders and White House officials have urged a muted and restrained approach. Show Dr. Blasey respect; offer to hear her out; and avoid questioning her credibility, at least directly, they have agreed in private conversations.

But many conservatives see little use in being deferential when, they argue, the Democrats play by no such rules. They look back at the failed confirmation of the Republican nominee Robert Bork in 1987, whose writings on civil rights were picked over by Democrats, and the 1991 hearings for Clarence Thomas, who faced testimony from Anita Hill that he had sexually harassed her, and they see a sophisticated and ruthless Democratic machine bent on discrediting their nominees.