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4 year oldn an interview, Tara Reade urged Mr Biden to "please step forward and be held accountable".
She added: "You should not be running on character for the president of the United States."
Mr Biden, who is set to challenge President Donald Trump in November, has denied Ms Reade's accusation.
WARNING: Some readers may find details in this story upsetting.
Ms Reade, now 56, worked as a staff assistant to Mr Biden from 1992-93 when he was a senator for the US state of Delaware.
She has said that in 1993 he forced her against a wall and put his hands under her shirt and skirt, penetrating her with a finger, after she delivered him his gym bag.
In her most graphic, detailed account yet of the alleged assault, Ms Reade told US media personality Megyn Kelly that Mr Biden kissed her neck and told her he wanted to have sex with her, using an obscene term.
"So, he had one hand underneath my shirt, and the other had, I had a skirt on, and he went down my skirt and then went up and I remember I was up almost on my tippy toes," she said.
"When he went inside the skirt, he was talking to me at the same time, and he was leaning into me and I pulled this way away from his head."
When she refused his advances, she said: "He looked at me and said, 'What the hell, man, I heard you liked me'."
She added: "He pointed his finger at me and he said 'You're nothing to me. You're nothing'."
Ms Reade was asked by Kelly if she wanted Mr Biden to withdraw from the race.
"I wish he would, but he won't, but I wish he would," she said. "That's how I feel emotionally."
Ms Reade offered to take a lie detector test about her claim, on condition that Mr Biden do so also.
"I will take one if Joe Biden takes one," she said.
Ms Reade said she had received a death threat after Biden supporters accused her without evidence of being a Russian agent.
"His surrogates have been saying really horrible things about me and to me on social media," she continued.
"He hasn't himself, but there is a measure of hypocrisy with the campaign saying it's been safe - it's not been safe.
"All of my social media has been hacked, all of my personal information has been dragged through."
Biden campaign communications director Kate Bedingfield said in a statement after the interview aired that Ms Reade's story contained "inconsistencies".
"Women must receive the benefit of the doubt," said the statement. "They must be able to come forward and share their stories without fear of retribution or harm - and we all have a responsibility to ensure that.
"At the same time, we can never sacrifice the truth. And the truth is that these allegations are false and that the material that has been presented to back them up, under scrutiny, keeps proving their falsity."
Mr Biden, who is the Democratic party's presumptive presidential nominee, broke his silence on the matter a week ago, appearing on a morning television show to brand the allegations "false".
A court document from 1996 shows Ms Reade's ex-husband describing "a problem she was having at work regarding sexual harassment, in US Senator Joe Biden's office", according to the San Luis Obispo Tribune.
"It was obvious that this event had a very traumatic effect on [Reade], and that she is still sensitive and effected [sic] by it today," wrote Theodore Dronen, her then-husband, in a legal memo during their divorce battle.
The file obtained by the California newspaper appears to be the only document from the time that might describe Ms Reade's allegation.
Ms Reade's brother, a former neighbour and a former colleague have all said they heard her describe the accusation against her boss after the alleged incident. And her mother appears to have called a CNN show about the claim back in 1993.
High-powered Manhattan lawyer Douglas Wigdor said in a statement that he is representing Ms Reade. He has represented alleged sexual assault victims of jailed Hollywood producer Harvey Weinstein.
Some Republicans are seizing on the Reade accusation to portray Democrats as hypocrites who only defend women when claims of wrongdoing are aimed at conservatives.
Mr Trump himself has faced a barrage of sexual misconduct allegations, which he also denies. He once boasted of grabbing women by the genitals.
However, the Democrats have much more strongly championed the #MeToo movement, which backs women who make accusations of sexual assault and calls for their stories to be heard.
Women are a core constituency for the party, traditionally giving more votes to Democratic candidates than Republicans.
Some liberal women have said they believe Tara Reade but will vote for Mr Biden anyway because they view Mr Trump as much worse.
November's presidential election will be the first of the #MeToo era, and Mr Biden has framed it as a "battle for the soul of America".
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