"As I told the committee during my hearing, a federal judge must be independent, not swayed by public or political pressure," he wrote. "That is the kind of judge I will always be.
"I will not be intimidated into withdrawing from this process. 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."
In the letter, Kavanaugh denied two allegations of sexual misconduct against him. The first, from Christine Blasey Ford, accused the judge of attempted sexual assault at a party during the early 1980s.
Ford made the allegation in a confidential letter sent to Feinstein and agreed to testify before the committee Thursday.
Ramirez said Kavanaugh exposed himself to her at a 1983 party while she and friends were playing a drinking game.
Kavanaugh denied the accusation.
"This alleged event from 35 years ago did not happen," he said in a statement. "The people who knew me then know that this did not happen and have said so. This is a smear, plain and simple."
Ramirez contacted several former classmates to ask if they recalled the incident and said she couldn't be sure it was Kavanaugh, The New York Times reported.
President Donald Trump again offered his support for Kavanaugh Monday, as Capitol Police arrested several protesters who demonstrated against the nomination to the high court.
Trump endorsed his appointee, calling him "a fine man" with "an unblemished past."
"These are highly unsubstantiated statements from people represented by lawyers," Trump told reporters at the United Nations General Assembly in New York City. "Judge Kavanaugh is an outstanding person and I am with him all the way.
"For people to come out of the woodwork from 36 years ago and 30 years ago and never mention it, all of a sudden it happens, in my opinion, it's totally political. It's totally political."
In Washington, dozens of protesters occupied the corridors outside the offices of Feinstein and Sen. Susan Collins, R-Maine, in the Dirksen Senate Office Building, to oppose Kavanaugh. Some wore black T-shirts with the slogan "Be A Hero," and several were arrested.
Collins is one of several Republicans on the Senate judiciary committee who have not publicly expressed approval for Kavanaugh.
Feinstein has called for an "immediate postponement" of Kavanaugh's scheduled confirmation vote Thursday.
In a letter to Grassley the chairman of the committee, Feinstein noted the two allegations.
"An investigation needs to be conducted as part of Judge Kavanaugh's background investigation by career professionals at the FBI -- not partisan staff of the committee," Feinstein wrote. "We need a fair, independent process that will gather all the facts, interview all the relevant witnesses, and ensure the committee receives a full and impartial report. Should the White House continue to refuse to direct the FBI to do its job, the committee must subpoena all relevant witnesses."
Feinstein, the top Democrat on the committee, added, "It is time to set politics aside. We must ensure that a thorough and fair investigation is conducted before moving forward."