Democrats are calling on Republicans on the Senate Judiciary Committee to delay a scheduled vote Thursday on Supreme Court nominee Brett Kavanaugh, after a woman accusing him of misconduct when they were in high school came forward.

All 10 Democrats on the Senate Judiciary Committee signed a letter dated Monday morning urging Grassley to postpone the committee vote.

"The lack of the nominee's full record, combined with his incomplete and misleading answers during the hearing and beyond, demonstrate a clear need for the committee to postpone moving forward this week," they wrote.

Get Breaking News Delivered to Your Inbox

Christine Blasey Ford's decision to go on the record Sunday in the Washington Post with the claims she wrote in a letter alleging Kavanaugh assaulted her in the 1980s at a party have intensified Kavanaugh's confirmation process. Although Senate Judiciary Committee Chairman Chuck Grassley plans to set up calls with both Ford and Kavanaugh, as of Sunday afternoon, the committee still intended to move forward with the vote Thursday. Grassley, as the top Republican on the committee, controls its schedule.

But some Republicans, too, expressed hesitation Sunday. Sens. Lindsey Graham, R-South Carolina, and Jeff Flake, R-Arizona, said they want to hear from Ford, and Sen. Bob Corker, R-Tennessee, told Politico it would be best for the committee to not vote on Kavanaugh until the committee hears Ford out.

On Monday morning, White House counselor Kellyanne Conway told reporters she, too, believes Kavanaugh should be heard — and Kavanaugh expressed a willingness to testify, through an adviser.

Many Democrats were already calling for the vote to be postponed on Sunday.

Senate Judiciary Committee Ranking Member Dianne Feinstein, whose office received the information in Ford's letter, said an FBI investigation should conclude before Kavanaugh's nomination proceeds. The FBI told CBS News' Jeff Pegues that the information was placed in Kavanaugh's background file, which is not the same thing as a separate investigation.

"I support Mrs. Ford's decision to share her story, and now that she has, it is in the hands of the FBI to conduct an investigation. This should happen before the Senate moves forward on this nominee," Feinstein said in a statement Sunday afternoon.

Supreme Court nominee Judge Brett Kavanaugh seen Tue., Sept. 4, 2018. Getty

Other Democrats on the committee had also expressed a desire to postpone the vote.

"Her story is very credible and I believe her ... This development is yet another reason not to rush Brett Kavanaugh's nomination. The committee should postpone this week's vote," Sen. Mazie Hirono, D-Hawaii, said.

Sen. Sheldon Whitehouse, D-Rhode Island, said Ford's move "requires a pause, at a minimum, in the unseemly, special-interest-funded rush to put Brett Kavanaugh on the court."

Sen. Amy Klobuchar, D-Minnesota, said she expects Ford's claims "to be seriously investigated by the FBI before we vote."

Senate Minority Leader Chuck Schumer, D-New York, who is not on the committee, said Grassley must postpone the vote until the allegations are thoroughly investigated, at the least.

On the House side, which does not take part in the confirmation of Supreme Court nominees, Speaker of the House Nancy Pelosi, D-Calif., said, "failure to postpone this vote without due process and a thorough investigation into these serious and credible allegations would be a dereliction of Congress' duty to demand zero tolerance of sexual harassment and abuse."

Ford, now a professor at Palo Alto University, told the Post she felt a "civic responsibility" to speak up about Kavanaugh. Ford told the Post she believes the alleged incident happened in 1982, when she was a 15-year-old sophomore at an all-girls school and Kavanaugh was 17 at an all-boys school. Ford described to the Post how she said Kavanaugh allegedly pinned her to the bed and groped her, attempting to remove her clothing. At one point when she tried to scream, Ford claimed, Kavanaugh allegedly held his hand over her mouth. Ford claimed she eventually escaped the situation.

Kavanaugh denied the allegation before Ford came forward, and the White House is standing by Kavanaugh's earlier denial. Kavanaugh issued another denial Monday morning.

"This is a completely false allegation. I have never done anything like what the accuser describes—to her or to anyone," Kavanaugh said in a statement. "Because this never happened, I had no idea who was making this accusation until she identified herself yesterday. I am willing to talk to the Senate Judiciary Committee in any way the committee deems appropriate to refute this false allegation, from 36 years ago, and defend my integrity."