The woman, Christine Blasey Ford, discussed her accusation in an interview with The Washington Post.
"I thought he might inadvertently kill me", the victim told the newspaper, adding, "He was trying to attack me and remove my clothing".
"She didn't say anything in our confidential session with Judge Kavanaugh".
At a high school house party, "He was trying to attack me and remove my clothing", Ford claimed, and said both Kavanaugh and a friend were extremely intoxicated during the incident, which happened when they were in high school.
Kennedy, who sits on the judiciary committee, told Fox News Sunday Democrats had done nothing during hearings to raise the contents of a letter about the alleged assault which was reportedly sent in July to the California Democrat Dianne Feinstein.
After the report was published Sunday, Senate Minority Leader Chuck Schumer, D-N.Y., called on Judiciary Committee Chairman Chuck Grassley, R-Iowa, to delay the confirmation vote "until, at a very minimum, these serious and credible allegations are thoroughly investigated", he wrote in a statement.
Mr Kavanaugh is strongly opposed by some Democrats for his views on issues such as abortion, and it seems unlikely any of them will cross the aisle to vote with their Republican colleagues.
After telling Eshoo's office, she lawyered up with attorney Debra Katz, who recommended that Ford take a lie detector test.
This came after elements of the allegations were made public in recent days, prior to the Washington Post interview.
If Kavanaugh had been accused by a single person and without additional evidence of cheating on an exam, stealing, or even of perpetrating a hit-and-run decades ago, one could imagine the nominee and his defenders explaining it away, changing the subject, and treating it as a distraction from getting him confirmed.
"She believes that but for his inebriation and his inability to take her clothes off, he would have raped her", Katz said. "Judge Kavanaugh and the White House both stand by that statement".
"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".
He said she used Kavanaugh's last name and voiced concern that he - then a federal judge - might one day be nominated to the Supreme Court.
"My biggest fear was, do I look like someone just attacked me?" she said. The senator wants the allegations to be heard promptly, she said. And whether or not such additional evidence is produced, there is no chance that Kavanaugh will be charged with a crime, since the statute of limitations has long expired, and he was a minor at the time of the incident.
Sixty-five women who knew Kavanagh in high school defended him in another letter, circulated by Senate Judiciary Committee Republicans, as someone who "always treated women with decency and respect".
Ford said that Kavanaugh forced himself on her by pinning her to a bed in the early 1980s, when the two were attending nearby high schools in Maryland.
Ford's husband told the Post that she used Kavanaugh's name in the therapy sessions and anxious he would be nominated to the Supreme Court. He called Roe an "important precedent" during his Senate Judiciary confirmation hearing but refused to say whether the ruling was correct.
The woman accusing Kavanaugh of sexual misconduct has not come forward publicly.
Now Ford has told The Washington Post she had chose to waive her anonymity because she felt her "civic responsibility" was "outweighing my anguish and terror about retaliation" after the basic outlines of the story emerged in media last week.