Rob Curtis told Buzzfeed news he felt like a "second-class citizen" when the session at renowned studio Sang Bleu, in Dalston, was cancelled.
Malvina Wisniewska denied prejudice and told the BBC she had previously tattooed clients with HIV.
It is illegal to discriminate against anyone based on their HIV status.
Mr Curtis, from Shoreditch, in east London, told the BBC he met with Ms Wisniewska twice before his appointment to discuss a complex geometrical tattoo on his thigh.
When he arrived at the studio to be inked on 28 January, the 36-year-old project manager indicated his HIV status on a disclosure form.
Shortly after, he was informed by a receptionist at the studio the work had been cancelled.
Ms Wisniewska said the disclosure had brought on a panic attack that left her feeling "vulnerable" and unable to complete the intricate work.
She had to undergo an emergency course of aggressive treatment after she came into contact with a sharp object while tattooing a different customer with HIV, she said.
During the "traumatic" six-month wait to be given the all clear, she said she became depressed.
She said: "On this occasion I felt very panicky, especially that Mr Curtis was the first customer since that time, who identified their positive status."
Mr Curtis said he felt "hurt" by her refusal.
While he empathised in part with her distressed mental state, he said the law was clear and he felt he had been discriminated against.
"At that point she was acting on stigma," he said.
"It's scary facing the prospect of HIV, let alone having it. I have some empathy with her but, if anything, because of that experience, she should have some compassion."
Sang Bleu studio said there should be no risk of HIV transmission to any artist if health and safety rules were followed. It has since dismissed Ms Wisniewska.
Its owner, Maxime Plescia-Buchi, said he was against discrimination and expected any artist working for him to share his ethics.
The Equality Act 2010 is designed to legally protect people from discrimination in the workplace and in wider society.
A tattooist who refused to work on a man with HIV has said a previous scare left her too "panicked" to complete the design.