Marouane Fellaini sorry for head-butt
While Fellaini went unpunished by the match officials at the time, replays clearly showed him making contact with Shawcross as they jostled for position in the box for a second-half corner.
Following the match, Toffees boss David Moyes said the incident was "not acceptable" and Fellaini later issued an apology on the club's official website.
"I apologise completely to Ryan Shawcross, my team-mates and to our fans at the game," said Fellaini. "There was a lot of pushing and pulling going on inside the Stoke penalty area and I didn't feel I was getting any protection from the officials. Nevertheless I know I shouldn't have done what I did.
"I have also apologised to the manager and the staff. I have no excuses, I was disappointed with the way I was being treated and I lost my temper, which was unprofessional of me."
Despite Fellaini's apology, the incident is likely to come under scrutiny from the Football Association's disciplinary panel when it reviews the footage this week.
Moyes said after the game that Everton would accept any punishment, with referee Mark Halsey's failure to act allowing the FA scope to hand down retrospective punishment.
Moyes said he thought Fellaini had warranted a red card and emphasized he would not stand for that sort of behavior from his players.
"We will take the punishment if it is given for it," Moyes said. "I have told (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, 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."
Stoke manager Tony Pulis said Fellaini had in fact been guilty of three unpunished offenses Saturday, and referenced others he felt had been committed by different players against his team this season.
"I'll say this -- David Luiz's challenge on Jon Walters (in a 1-0 defeat at Chelsea), Ciaran Clark's challenge on Glenn Whelan last week (in a 0-0 draw at Aston Villa), and then the three incidents today," Pulis said.
"I thought the rules were set out in stone. I like Mark Halsey and I think he's a good referee, but I don't know whether he has seen it. If he has seen it and not acted, then you really want to know why he has not acted. I don't think you can hide behind anything and what (Fellaini) has done is wrong."