The charge followed an incident during Stoke’s 1-0 win over Arsenal at the Britannia on Saturday, when Adam tangled with Gunners striker Olivier Giroud.
Adam was alleged to have stamped on the striker’s leg, an incident that was missed by referee Mike Jones and his officials at the time.
But having reviewed the footage on video, the Scottish midfielder was charged and found guilty at an Independent Regulatory Commission hearing.
He will now miss the games against Norwich, West Ham and Aston Villa.
Stoke City chairman Peter Coates described the FA’s decision to charge Adam as “appalling” on Tuesday and the club’s chief executive Tony Scholes was also furious with the FA after the appeal was rejected.
“I am extremely disappointed for the effect that this is going to have on Charlie and on the football club,” Scholes told the club’s official website.
“However, I am not surprised because clubs like ours very rarely succeed in appeals to the FA.
“We put together a strong defence which was based upon the facts, which were that Charlie needed to plant his foot, and in changing direction his eyes followed the ball at all times.
“How can anybody other than Charlie say what intent was there? How can the FA possibly justify this decision in the light of recent incidents which were far worse at other clubs, which have gone unpunished?”