The Belgium international appeared to headbutt Potters defender Ryan Shawcross during the second half at the Britannia Stadium and he is now likely to be charged when FA officials review his actions on Monday.
When asked if Fellaini had overstepped the mark, Moyes replied: "Yes he did and we will accept whatever punishment is given out for it.
"I have told him in the dressing room that it is not acceptable and whatever gets handed out he and us deserve it."
Moyes 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 had been able to see it, it should have been."
A 36th-minute own goal from Shawcross put Everton ahead but Kenwyne Jones levelled in the 52nd minute after Leon Osman had missed a glorious chance to net a second for the visitors.
Moyes said: "It was a really hard-fought game. We got in front but it was tough. We didn't play well today - as well as we have been playing that's for sure - and we made mistakes.
"I think everybody will find it hard here. But we got a goal up - although I wouldn't have said we necessarily deserved to be a goal up - and then we miss a great chance.
"We then had a little bit of good football that should have put us two up but it didn't. Then, what you would expect when you come to Stoke, they rode right back into it and got a goal, albeit Tim (Howard) should have saved it. Then we were under quite a bit of pressure after that."