Adam is alleged to have stood on Arsenal's Olivier Giroud in the second half of the club's 1-0 win over the Gunners on Saturday in an incident missed by referee Mike Jones.
Adam faces a three-match ban if found guilty and despite the possibility of that punishment being increased, Coates has declared the club's intention to appeal "vigorously".
Coates told the Stoke Sentinel: "We feel this is trial by media. We are disgusted.
"Charlie absolutely assures us it was not intentional. We will defend it vigorously. It is appalling."
Coates is said to be particularly aggrieved that other recent incidents that were missed at the time by match officials went unpunished.
Coates added: "We have seen some of the things that have gone on in recent weeks and no-one has been called to account."