Richard O'Dwyer created the TVShack.net website, which did not host any copyrighted material but linked to other sites that did.
The 24-year-old was arrested over the site in 2011, but will not face any prosecution in the UK.
However, the US has stated that because some American citizens used the site, he must be extradited to face criminal charges around copyright infringement, which could see him spend up to 10 years in a US prison.
Wales launched the Change.org petition on Sunday (June 24), which has already amassed over 12,000 signatures.
The petition personally targets Prime Minister David Cameron, deputy PM Nick Clegg and home secretary Theresa May, who signed the extradition order in March.
He described O'Dwyer as "the human face of the battle between the content industry and the interests of the general public".
Wales told The Guardian: "When I met Richard (along with his mother), he struck me as a clean-cut, geeky kid. Still a university student, he is precisely the kind of person one can imagine launching the next big thing on the internet.
"Enthusiastic, with a sharp mind and a quick wit, he reminds me of many great entrepreneurs. He tried to follow the law, and I would argue that he very likely succeeded in doing so."
He added: "Given the thin case against him, it is an outrage that he is being extradited to the US to face felony charges for something that he is not being prosecuted for here. No US citizen has ever been brought to the UK for alleged criminal activity that took place on US soil."
Meanwhile, O'Dwyer said that he is coping "quite well" about his threat of extradition to the US.
"It does get in the way, it distracts you... if you thought about extradition all day you'd never get any work done. It'd be a horrible mess," he told The Guardian.
"It's quite difficult but I think I'm managing quite well. I think about it sometimes during the day, but I try to think about other things that are more important. I don't let their extradition warrant ruin my life.
"Otherwise you'd fail university, just sit in your room all day moaning. They'd be winning if I let it do that."
A Home Office spokesman said: "We have effective, fair and balanced extradition arrangements with the US and other international partners.
"People who have committed serious offences such as murder, rape, other sex crimes and fraud, have been successfully extradited to the UK and convicted.
"It should also be noted that our courts have refused to extradite nine people requested by the US since 2004, while US courts have not refused any of our extradition requests."