US President Donald Trump's Supreme Court nominee Brett Kavanaugh on Monday said he will "not be intimidated into withdrawing" his nomination after a second woman came forward with a sexual misconduct allegation against him.

"As I told the Committee during my hearing, a federal judge must be independent, not swayed by public or political pressure.  That is the kind of judge I will always be. I will not be intimidated into withdrawing from this process,” Kavanaugh said in the letter.

"The coordinated effort to destroy my good name will not drive me out. The vile threats of violence against my family will not drive me out. The last minute character assassination will not succeed,” Kavanaugh said and refuted a new accusation that he exposed himself to a classmate at Yale University, also in the 1980s.

Justice Kavanaugh already faces a separate allegation of sexual assault made by Christine Blasey Ford, a professor at Palo Alto University, who claims Kavanaugh attempted to sexually assault her at a drunken high school party in the early 1980s.

Ford and Kavanaugh are set to testify before the Senate Judiciary Committee on Thursday to address the allegations after a week of fraught public negotiations between Ford's attorneys and Senate Republicans.

The latest sexual assault allegation dated back to the 1983-84 academic school year, when Justice Kavanaugh was a freshman at Yale University, The New Yorker said, adding that at least two Senate Democrats were investigating the allegation.

Kavanaugh has denied both the allegations. “These are smears, pure and simple. And they debase our public discourse. But they are also a threat to any man or woman who wishes to serve our country.

"Such grotesque and obvious character assassination—if allowed to succeed—will dissuade competent and good people of all political persuasions from service,” he said in the letter, a copy of which was released by the White House.

Senate Judiciary Committee has scheduled a new hearing for him and the first accuser. By the time first allegation surfaced, he had testified before the Senate Judiciary Committee for more than 31 hours.

"Only after that exhaustive process was complete did I learn, through the news media, about a 36year-old allegation from high school that had been asserted months earlier and withheld from me throughout the hearing process. First it was an anonymous allegation that I categorically and unequivocally denied,” he said.

“Soon after the accuser was identified, I repeated my denial on the record and made clear that I wished to appear before the Committee. I then repeated my denial to Committee investigators—under criminal penalties for false statements,” he said.

Kavanaugh said all of the witnesses identified by Ford as being present at the party she describes are on the record to the Committee saying they have no recollection of any such party happening.

"I asked to testify before the Committee again under oath as soon as possible, so that both Dr. Ford and I could both be heard.  I thank Chairman Grassley for scheduling that hearing for Thursday,” he said.

“Last night, another false and uncorroborated accusation from 35 years ago was published.  Once again, those alleged to have been witnesses to the event deny it ever happened.  There is now a frenzy to come up with something—anything—that will block this process and a vote on my confirmation from occurring,” Kavanaugh said.