Ivory Coast players Eboue and Drogba, who was instrumental in Chelsea's run to the Champions League title last season, were heckled during Sunday's 2-1 league win by Fenerbahce in the high-octane encounter.
A Fenerbahce fan was stabbed to death and more than 100 fans were detained following Turkey's most popular derby.
Drogba, whose Galatasaray team have already secured the Turkish championship, wasted little time in highlighting that both Fenerbahce goals were scored by Cameroon striker Pierre Webo.
"You call me monkey but you cried when Chelsea beat Fenerbahce in 2008, you called me monkey but you jumped in front of your screen when I won the Champions League," Drogba said on Galatasaray's Facebook page.
"You called me monkey but you got mad when I became champion with Galatasaray and the saddest thing is you called me monkey and forgot that you jumped when my 'monkey' brother scored twice yesterday... and you call yourself a true fan? Check all the Galatasaray fans comments and learn from them."
Drogba also posted an image on the online photo-sharing network Instagram depicting humans evolving from apes.
A man claiming to be the supporter who waved the banana told a local television show that he had the fruit in his hand to eat but declined to give his name.
"I didn't wave the banana with the aim of racism, I just joined a jeering against [Galatasaray goalkeeper Fernando] Muslera," the man said during a phone-in.
"I am not like that. I apologise to Fenerbahce and Galatasaray fans," he added.
