“I told him that given that this issue could possibly come before the court, if the justice was to vote a certain way, what that would mean for millions of people, what that would mean for me on a personal basis, what that would mean for my dad,” said one of those in the meeting, Jacob Redman, a 22-year-old college senior.

Mr. Redman lost his health insurance when his father, who has a genetic condition that causes tumors on the spine, lost his job; the two men turned to Medicaid, which has been expanded in West Virginia under the Affordable Care Act. He said Mr. Manchin told him he was “still on the fence” about Judge Kavanaugh.

Senate Democrats have been avoiding the standard meet-and-greet courtesy visits with the judge as they clash with Republicans over access to emails from his time as staff secretary to former President George W. Bush. But Democrats appear to be losing that battle; on Friday, Republicans asked the National Archives for certain documents related to Judge Kavanaugh, but they left the Bush White House emails out of the request.

“What are they hiding?” Senator Chuck Schumer of New York, the Democratic leader, asked on Monday. “What are they afraid of? Why wouldn’t they normally grant the kind of openness to records that America prides itself on?”

Even as lawmakers argued about the release of documents, however, the National Archives on Monday began releasing some files related to Mr. Kavanaugh’s time as a lawyer working for the independent counsel Kenneth W. Starr’s Whitewater investigation of President Bill Clinton. Those documents — 1,025 pages of files — were released in response to Freedom of Information Act requests from news organizations, including The New York Times.