The England international unsuccessfully argued that the punishment was excessive but the FA's independent panel rules him out of matches against Chelsea and West Ham
As the north Londoners were beaten 6-3 at the Etihad Stadium, the England international was caught on camera appearing to extend his middle finger towards supporters.
Deemed a "sending-off offence" by the Football Association's three-person panel covering events not seen by officials during matches, the gesture has cost Wilshere his eligibility for Monday's Premier League encounter with Chelsea.
He will also miss the Gunners' Boxing Day clash with West Ham.
"Whilst admitting the charge, Wilshere claimed the standard sanction for this offence was clearly excessive," said a statement on the FA's official website.
"The [Independent Regulatory] Commission rejected this claim and the two-match suspension will commence with immediate effect."