Fox was red carded in the 81st minute but Chris Coleman told the CT: "I have seen it again and looking at the video evidence it is quite difficult.
"Danny was a bit disappointed and I think it was heat of the moment out of frustration at what had happened after being 2-0 up. He just swung his elbow and caught the guy in the midriff."
Based on this, the club have decided not to appeal and Fox willl miss City's game at Ipswich on Sunday and the first two league games of next season.