Lawyers for Christine Blasey Ford say she has “been the target of vicious harassment” in the wake of her claims and has been forced to leave her home.
The clinical psychology professor had been asked to give evidence at a Senate panel hearing which was scheduled for Monday.
But in a letter to the panel’s chairman Chuck Grassley, her lawyers said “a full investigation by law enforcement officials” was needed first to ensure the “crucial facts and witnesses in this matter are assessed in a non-partisan manner”.
Dr Ford has accused Judge Kavanaugh – Donald Trump’s Supreme Court nominee – of pinning her to a bed at a party in Maryland in the early 1980s when she was 15.
She alleges that the then-17-year-old tried to remove her clothing and put his hand over her mouth when she tried to scream.
“She believes that if were not for the severe intoxication of Brett Kavanaugh, she would have been raped,” her lawyer Debra S Katz told NBC.
Judge Kavanaugh, 53, has called the allegation “completely false”, while Mr Trump has voiced sympathy for him.
“I feel so badly for him that he’s going through this, to be honest with you,” the US president told a news conference.
“This is not a man that deserves this. Hopefully the woman will come forward, state her case. He will state his case before representatives of the United States Senate. And then they will vote.”
Sixty five women who knew Judge Kavanaugh in high school have defended him in a letter, calling him someone who “always treated women with decency and respect”.
Meanwhile, Mr Trump’s eldest son appeared to mock Dr Ford’s sexual assault allegation.
Donald Trump Jr posted an image on Instagram on Monday with the caption “Judge Kavanaugh sexual assault letter found by Dems…”
The photo attached shows a crumpled-up piece of notebook paper with a scribbled message: “Hi Cindy will you be my girlfriend, Love Bret.”
The note has boxes to check for “yes” or “no” and seems to compare Judge Kavanaugh’s accuser to a playground crush.
Judge Kavanaugh appeared to be on course to become a Supreme Court Judge but Dr Ford’s allegations have disrupted the process.
The accusation has raised the prospect of congressional Republicans defending Mr Trump’s nominee ahead of mid-term elections.
Critics have already accused the party of fast-tracking the process to get Judge Kavanaugh on the court by 1 October, the first day of the autumn term.