The former Chelsea striker joined the Turkish outfit from Shanghai Shenhua late last month in a controversial move as he was still contracted to the Chinese club.
Schalke managed to draw 1-1 with Galatasaray in their Champions League round of 16 first leg on Wednesday night but that did not stop the club launching a protest regarding the eligibility of Drogba playing for the Istanbul giants.
"We decided after a thorough legal advice to take this step," general manager Horst Heldt said in a statement on the club's official website.
"It is our duty to examine their eligibility to play Drogba."
Schalke's chief financial officer Peter Peters added: "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."