Both the Premier League champions and midfielder Barry, 31, said they would "reluctantly" accept the punishment.
Barry admitted an FA charge of using abusive and or insulting words to an official after the 3-2 defeat at the Etihad, which was sealed by a late Robin van Persie free-kick.
But City have written to the FA to voice their disappointment that Barry was charged for what they felt was normal behaviour in a heated situation.
A club statement said: "Both the club and Gareth Barry reluctantly accept the punishment, which leads to Gareth being unavailable to take part in the weekend's match against Newcastle."
The FA said Barry had admitted "a breach of FA Rule E3 in that he used abusive and or insulting words towards a match official following his side's match against Manchester United on Sunday 9 December 2012".