TV cameras caught the Stoke City midfielder standing on Giroud's leg after the ball had been played.
During the second half of the Gunners' 1-0 defeat at the Britannia Stadium on Saturday, television cameras showed the Scotland midfielder standing on the leg of the French striker after he had passed the ball.
An FA statement read: "The incident, which was not seen by the match officials but caught on video, occurred during the fixture between the two sides at the Britannia Stadium on Saturday March 1 2014. Adam has until 6pm on Tuesday 4 March 2014 to respond to the charge.
"Under a new pilot project in Premier League matches this season, if an incident has not been seen by the match officials, a three-man panel of former elite referees will be asked by The FA to review it and advise what, if any action, they believe the match referee should have taken had it been witnessed at the time.
"For an FA charge to follow, all three panel members must agree it is a sending-off offence. In this instance, the panel were of the unanimous decision that it was an act of violent conduct."
Stoke says it is "surprised and disappointed" by the FA's judgement and confirmed the club will appeal "vigorously" against the decision.