The Duke of York has held the Earl of Inverness title since his marriage to Sarah Ferguson in 1986.
The Prince 'categorically denies' the allegation.
Mr Davies said: "I think that him holding the title of Earl of Inverness puts a stain on our beautiful city.
"I just feel it is embarrassing to have someone like Prince Andrew associated with Inverness."
He added: "I started the petition despite being a huge fan of the Royal Family, especially the Queen, who I think has been wonderful over the past 70 years."
Prince Andrew announced earlier this week that he will be stepping back from public duties after his television interview, in which he appeared to show little empathy for Epstein's victims or remorse for his friendship with the sex offender, was widely derided.
In the wake of the broadcast, a number of charities have cut ties with the Prince, including Huddersfield University.
BT said on Wednesday it would no longer work with iDEA – the Duke of York Inspiring Digital Enterprise Award – while the duke was a patron of the scheme.
Meanwhile, the accountancy firm KPMG, a founding partner of the [email protected] entrepreneurial scheme devised by the Prince, said that its sponsorship contract had expired at the end of October and would not be renewed.
Banking services company Standard Chartered said it would not be renewing its sponsorship of [email protected] – which claims to have generated more than 6,000 jobs – in December, citing “commercial reasons”.