Andrew is facing fresh scrutiny after Virginia Giuffre's first UK television interview was aired as part of a BBC Panorama investigation into the duke's association with convicted sex offender Jeffrey Epstein.
Ms Giuffre claims she was trafficked to London by Epstein and forced to sleep with the duke when she was a teenager, which Andrew categorically denies.
She spoke of how she was left "horrified and ashamed" after the alleged encounter in 2001, saying: "One thing that I can tell you is you never forget the face of someone who has heaved over you."
She claimed she danced with Andrew in Tramp night club, adding he was "the most hideous dancer I've ever seen in my life" and "his sweat was ... raining basically everywhere".
Her father, Sky Roberts, told ITV's Good Morning Britain: "I know she's telling the truth. My daughter is, she's a good girl, she really is."
He added: "I feel really sorry that all this happened, but it did happen and Prince Andrew should pay for it."
Mr Roberts expressed his support for Ms Giuffre.
"At the time she was going through it, I had no idea," he said.
The royal family will be out in force at Buckingham Palace today, at a reception hosted by the Queen for Nato leaders including US President Donald Trump, but Andrew will be missing, having quit public life in the wake of his disastrous Newsnight interview in which he sought to address Ms Giuffre's claims.
Mr Trump said at the US ambassador's residence when asked about the duke stepping down from royal duties: "I don't know Prince Andrew, but that's a tough story, it's a very tough story."
A Buckingham Palace spokesman said in response to the allegations on Panorama: "It is emphatically denied that the Duke of York had any form of sexual contact or relationship with Virginia Roberts. Any claim to the contrary is false and without foundation."
Panorama disclosed that five women who allege they were victims of paedophile Epstein are calling on Andrew to give evidence in their civil court case.
They claim the duke saw people being given massages at Epstein's homes.
The Queen's second son has insisted he did not witness or suspect any suspicious behaviour during his visits to Epstein's homes in Florida, New York and the Caribbean.
Panorama claimed it had also uncovered an email allegedly sent by Andrew in 2015 to Ghislaine Maxwell, who is accused of procuring girls for Epstein.
In it, the duke is reported to have asked for Ms Maxwell's help in dealing with claims by Ms Giuffre, who was previously known as Virginia Roberts.
The duke is said to have written: "Let me know when we can talk. Got some specific questions to ask you about Virginia Roberts."
Ms Maxwell replies: "Have some info. Call me when you have a moment."
The programme also examined the well-known photograph of Andrew pictured with his arm around Ms Giuffre's waist at Ms Maxwell's house.
The duke has said he had no memory of the photo being taken and questioned whether it was his own hand in the image.
But Ms Giuffre insisted the photo was genuine and said she gave the original to the FBI in 2011.
In his recent Newsnight interview, Andrew claimed he had a medical condition in 2001, after suffering an overdose of adrenaline in the Falklands War when he was shot at, which meant he did not sweat.
He also denied having sex with Ms Giuffre, saying: "I can absolutely categorically tell you it never happened", adding: "I have no recollection of ever meeting this lady, none whatsoever."
Ms Giuffre claimed in court papers in Florida that she was forced to have sex with Andrew when she was 17, below the state's age of consent.
She alleges that the duke slept with her on three separate occasions, twice while she was under age.
During his Newsnight interview, the duke said an alleged encounter with Ms Giuffre in 2001 did not happen as he spent the day with his daughter, Princess Beatrice, taking her to Pizza Express in Woking for a party.
Epstein took his own life in a New York prison in August while he was being held on sex trafficking charges.