The Juventus coach saw his side crash out of the competition 1-0 in a fixture that was halted after 31 minutes on Tuesday due to snow storms.
After returning to a slow and muddy pitch a day later, Wesley Sneijder scored the all-important winner five minutes from time to move Galatasaray second in the group, behind Real Madrid, and condemn Juventus to UEFA Europa League football in the new year.
Conte was clear that the game should not have taken place on Wednesday, describing the match as a "football brawl".
"We certainly tried to have the game postponed and met with the UEFA delegate beforehand, but nobody wanted to listen," Conte told Sky Sport Italia.
"(Galatasaray coach) Roberto Mancini said it was dangerous and I agreed with him.
"At half-time I reiterated to the referee that it was not safe to play in these conditions. Didier Drogba said it was the same thing for both teams, but I pointed out it wasn't, because we were trying to play football.
"I have to praise the lads, as they gave their all in a situation where we were enormously penalised, as the game resorted to a football brawl.
"We couldn’t have predicted these circumstances, so in my view our exit is unfair, but we also made life difficult for ourselves."
Sneijder's goal was the first Juventus had conceded in three matches and the Netherlands international's fourth goal in his last four home outings.
