Fellaini is facing the prospect of retrospective action from the Football Association over the incident in the second half - seemingly missed by the match officials - where he appeared to thrust his head into the face of Potters captain Shawcross as the two men jostled with each other for position in the hosts' box.
Moyes said after the game that Everton would accept any punishment for Fellaini's actions, admitting he thought they had warranted a red card and emphasising that he would not stand for that sort of thing from his players.
"We will take the punishment if it is given for it," Moyes said.
"I have told him (Fellaini) in the dressing room that it is not acceptable - I won't accept it as a manager that he does that."
Asked how he might compensate for losing a player as influential as Fellaini to suspension should the 25-year-old be handed a ban by the FA, Moyes said: "There is no absence yet, but he is a really good player and a big part of the team, and that is why I think the biggest disappointment, if anything does happen, is that he will have let the team down."
He added: "I only normally complain if I think we are getting hard done by and I have said a few things about how I think we have been treated by decisions recently.
"So I think if you are going to talk about it with your own team... Today [Saturday], it should have been a sending-off in the game.
"If the referee (Mark Halsey) had been able to see it, it should have been."