The former director of international relations at the world governing body had hoped to succeed Sepp Blatter later this year, but failed to meet the requirements to run for office.
Under FIFA rules, candidates must have secured the required five letters of nomination by January 29, but Champagne has only been able to acquire three.
The Frenchman made the announcement in an open letter, which also criticised the current regime for not allowing a fair election process.
"I regret to announce that I have not presented the five sponsorship letters needed to be registered as a candidate in the election of 29 May," he said.
"I warmly thank the three federations that have endorsed me and the many presidents who explained with candour and friendship, that they could not do it despite their interest in my program. The reasons were numerous.
"Because they feared reprisals from their confederations having issued 'recommendations'. Because their federations were candidates to host continental competitions.
"Because they relied too heavily on the financial support. Because they were committed to defend a united continental front.
"[And] because some of the presidents were themselves engaged in an election or simply preferred another candidate.
"The institutions have mobilized to eliminate the only independent candidate. The latest events orchestrated in secret with barely veiled intentions by one of them, distributing letters of support between candidates, made me lose sponsorships especially in Europe.
"I also note that I would have been a candidate with the old version of the rules."
Champagne's decision leaves Prince Bin Al Hussein of Jordan, Dutch Football Association president Michael van Praag and former player Luis Figo in the running alongside the incumbent Blatter.
