Midfielder Fabregas was caught by Arteta's arm six minutes before the end of the Gunners' 4-1 defeat of Everton and flung himself to the ground clutching his face.
Moyes was furious and the home fans chanted "cheat, cheat" at the 20 year-old as referee Martin Atkinson showed Arteta the red card.
Moyes' after-match condemnation of Fabregas was laced with sarcasm as he said: "Fabregas obviously has a broken jaw or he would not have rolled around like he did."
Bendtner had already been booked and was shown only a second yellow card for the lunge at Johnson.
Moyes said: "I don't make the rules, it was probably the wrong decision by the referee to show a yellow for (Bendtner's) tackle, it should have been a straight red.
"It doesn't help our case, but it's over now. I can't do anything about it.
"I have said since the derby (with Liverpool in October) that I will not get involved too much with referees, we will just have to take it.
"But compare the incidents. We lose Mikel for three games while Bendtner only misses one for two yellows, you should see the state of Johnson's shin.
"We won't be appealing, you have seen what happens to everyone else when they appeal these days."