Austria teammate David Alaba seemingly tried to silence Arnautovic by grabbing hold of his jaw during the angry celebration.
“There were some heated words yesterday in the emotions of the game for which I would like to apologize, especially to my friends from North Macedonia and Albania,” he wrote.
“I would like to say one thing very clearly: I AM NOT A RACIST!
“I have friends in almost every country and I stand for diversity. Everyone who knows me is aware of that.”
European football’s governing body opened an investigation into the incident Tuesday and announced the suspension on Wednesday.
Arnautovic will now miss his nation’s match against the Netherlands on Thursday but will be available for the final group game against Ukraine on Monday.