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5 year oldLucy Flores, a former Nevada Assembly member, says Mr Biden kissed her on the back of her head at a campaign event.
"I had never experienced anything so blatantly inappropriate," she wrote.
Mr Biden said he had shown "expressions of affection" to people over the years, adding "not once - never - did I believe I acted inappropriately".
"If it is suggested I did so, I will listen respectfully," he said.
Serving as Barack Obama's vice-president for eight years and in the US Senate for nearly four decades, Mr Biden is seen as a possible frontrunner in the race for the Democratic presidential nomination even though he has not announced his candidacy.
The allegations first appeared on Friday in an article Ms Flores wrote for The Cut magazine.
Ms Flores was running as the Democratic candidate for Nevada's lieutenant governor in 2014 when the then-vice president flew in to support her bid.
As she prepared to go on stage, Ms Flores say Mr Biden placed two hands on her shoulders from behind.
"I felt him get closer to me from behind. He leaned further in and inhaled my hair. I was mortified," she wrote. "He proceeded to plant a big slow kiss on the back of my head. My brain couldn't process what was happening."
Mr Biden's spokesman Bill Russo initially said neither the vice-president nor his staff had any idea that "Ms Flores had been at any time uncomfortable, nor do they recall what she describes".
But on Sunday Mr Biden issued a statement himself, and promised to "pay attention" to Ms Flores - while reiterating he did not remember the incident.
"In my many years on the campaign trail and in public life, I have offered countless handshakes, hugs, expressions of affection, support and comfort. And not once - never - did I believe I acted inappropriately," it read.
"But we have arrived at an important time when women feel they can and should relate their experiences, and men should pay attention. And I will," he added.
Speaking in a CNN interview on Sunday, Ms Flores said Mr Biden's statement was "certainly better" than his spokesman's initial response, but called his behaviour "completely inappropriate" and something to think about of a person "who is considering running for president".
A number of candidates for the 2020 Democratic presidential nomination have backed Ms Flores.
"I think what this speaks to is the need to fundamentally change the culture of this country," Senator Bernie Sanders told Face the Nation on CBS, saying he had no reason not to believe Ms Flores.
She previously campaigned for Mr Sanders in his 2016 nomination run.
Senator Elizabeth Warren said Mr Biden "needs to give an answer", and Senator Amy Klobuchar said that in politics "people raise issues and they have to address them".
But many supporters have leapt to Mr Biden's defence.
Cynthia Hogan, a former aide to the vice-president, told the New York Times that Mr Biden "treated us with respect and insisted that others do the same".
A photo of Mr Biden holding her shoulders during her husband's swearing-in ceremony drew questions at the time and afterwards.
But in a blog post, Ms Carter wrote that the vice-president "could sense I was uncharacteristically nervous" at the event "and quickly gave me a hug", she wrote.
"But a still shot taken from a video... came to be the lasting image of that day."
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