Last Friday, the tabloid paper printed the photos - thought to have been taken on a camera phone while the Prince was on a private break with friends in Las Vegas - claiming that it was in the public interest.
US gossip website TMZ was first to publish the images and then they were widely distributed across the internet. The two pictures included one of the Prince with a naked woman in a hotel room.
The PCC has said that all the complaints against The Sun were about invasion of privacy and they came from members of the public, and not the royal family.
The watchdog said that it would consider the complaints and see whether it should take further action.
St James's Palace had previously requested that the photos not be printed in the British press over concerns that the move could infringe the Prince's right to a private life.
David Dinsmore, managing editor of The Sun, said that the paper had thought "long and hard" about publication, but judged that the case was "about the freedom of the press".
He added: "This is about the ludicrous situation where a picture can be seen by hundreds of millions of people around the world on the internet but can't be seen in the nation's favourite paper read by eight million people every day.
"This is about our readers getting involved in the discussion with the man who is third in line to the throne - it's as simple as that."
Rupert Murdoch, the boss of media giant News Corporation, backed the decision and said that that there is "no such thing as a free press in the UK", as the "internet makes a mockery of these issues".
Despite racking up 3,600 complaints, the Prince Harry photos lag far behind the record for controversial newspaper articles.
In November 2009, a total of 25,000 complaints were lodged with the PCC over a story by Daily Mail columnist Jan Moir about the death of former Boyzone singer Stephen Gately, on grounds that it was homophobic.
> The Sun sparks media debate with Prince Harry photos
> Charlotte Church criticises The Sun over Prince Harry naked photos
Prince Harry's wild antics so far - pictures:
Copyright: Rex Features Paul Grover/Rex Features