The Football Association handed Wilshere a two-match for the gesture, which came during the second half of the Gunners' 6-3 defeat at the Etihad Stadium.
The England midfielder missed Monday's 0-0 draw against Chelsea and he will also sit out the Boxing Day clash at West Ham.
Wilshere, who has two children, admits he made a mistake in getting riled by the supporters who abused him, and he is braced for more barracking from the stands in the future.
Wilshere tweeted: "Gutted not to be Involved last night! Shouldn't of reacted the way I did but I know all you dads out there love your kids the way I do....
"Suppose I will just have to take the abuse about my kids in future......anyway, one more game to go! Can't wait to be back! #AFC"
Arsenal missed Wilshere on Monday evening.
The north London club looked flat and lacked a cutting edge against a well-drilled Chelsea team who sat back and showed little attacking intent.
The football on show was dull for large parts, and the main talking points from the match centred on refereeing decisions and late sliding tackles.
John Obi Mikel and Ramires were lucky to stay on the field after committing rash fouls on Mikel Arteta while Branislav Ivanovic and Mesut Ozil clashed after the former raised his studs perilously close to the German's face.
Theo Walcott was denied a penalty when he was tripped in the box by Willian - something which caused the Arsenal winger to run over and plead loudly to referee Mike Dean.
Walcott was happy to take a point from the game, especially as the blustery conditions in north London were extremely tough to play in.
"When you want to win titles, these are the games you can't lose," Walcott said. "Chelsea came with a negative attitude, but they did a job.
"We couldn't deal with the weather conditions, it was probably the craziest game I have ever been involved in.
"Considering the conditions I'm surprised everyone stayed on the pitch.
"It was the wind, I have never known anything like it. It was very difficult to judge everything. I could tell that both sets of players were struggling to get hold of the ball and slow the tempo down."
Although Arsenal missed the chance to move back to the top of the Premier League, Walcott feels the team stand a good chance of reaching the summit again soon.
He added: "It's so close, you can't really call it this year and it will go right down to the wire. We just have to make sure we win our next few games now.
"It will be a very good point come the end of this Christmas period."
Arsenal are braced for another intense physical clash against West Ham.
The Hammers have only beaten their neighbours at Upton Park once in the last 14 years.
But with West Ham just one point above the relegation zone, Walcott knows the east Londoners will not be easy opposition.
"It's going to be a good battle," the 24-year-old said.
"They had a good result against Tottenham in the Capital One Cup, we have to make sure we're ready.
"We'll make sure we recover well. In this period we can't train as much, so we have to make sure we sleep, recover and drink well."
Arsene Wenger will rotate his team for the London derby, but he is reluctant to throw Lukas Podolski back in to the starting XI given that the striker has been out for the last four months because of a torn hamstring.
"Podolski has not played a game at all since the second game of the season," the Arsenal manager said.
"He lacks a bit of competitiveness. He can only find that if I play him. But he has just two weeks' training.
"I have to make that decision whether I use him or not."