The German side took a creditable 1-1 draw away from the first leg of the European tie but have raised questions over Gala's use of the former Chelsea striker.
Drogba signed for Galatasaray at the end of January after ending his short stay with Chinese side Shanghai Shenhua amid claims he had not been paid his salary.
Shenhua countered that Drogba was in breach of his contract and it fell to FIFA to provisionally clear the way for the Ivorian to make his Galatasaray debut last week.
Schalke nevertheless have misgivings about Drogba's participation last night and contacted UEFA in a bid for clarification.
"We decided after a thorough legal advice to take this step," general manager Horst Heldt said in a statement on club website schalke04.de. "It is our duty to examine their eligibility to play Drogba."
Drogba agreed to join Galatasaray on January 28, three days before UEFA's deadline relating to the registration of players for the last 16 of the Champions League.
The contract wrangle with Shenhua meant Drogba did not receive international clearance from FIFA to make his Galatasaray debut until February 13.
Article 18.18 of UEFA's Champions League regulations states that the February 1 deadline for registering players for the knockout stage cannot be extended.
Schalke's chief financial officer Peter Peters said: "There is the suspicion that he received his eligibility after the end of the deadline to register the player for the UEFA Champions League.
"Before each game UEFA states that every club has to take responsibility for the players used.
"It is now up to UEFA to decide and Schalke will respect this process," said Peters.
It is understood UEFA have received Schalke's complaint, which will be passed to the organisation's disciplinary panel for investigation.
Poor losers, after drawing their tie 1:1 :-p Lol. He got done over in China, Gala have every right to sign and register Drogba to play in Europe! Gala are a big club, and a regular competitor in European competition. I'm sure they know the rules regarding player registration, and a club of their stature would surely not field a player who was ineligible or illegally registered?! Even though it was a 1:1 draw, sounds like sour grapes on the part of Schalke.- cleggy (nufc)