The 28-year-old has been charged by English football's governing body for stamping on Olivier Giroud during the Potters' 1-0 win against Arsenal at the Britannia Stadium in the Premier League on Saturday.
The incident was not seen by official Mike Jones and his assistants, but it has now been reviewed by a panel of three former referees.
Adam has until 6pm on Tuesday to appeal against the charge, and club chairman Coates insists that they will do so "vigorously".
"We feel this is trial by media," fumed Coates. "We are disgusted.
"We have seen some of the things that have gone on in recent weeks and no one has been called to account.
"Charlie absolutely assures us it was not intentional. We will defend it vigorously. It is appalling."