But Moyes said: "Cesc questioned Everton's integrity by suggesting we had possibly given money to the ref.
"That's why I felt if I or any player had said that, I think it would have warranted a sending off."
Arsenal skipper Fabregas was angered by Everton striker Louis Saha's 24th-minute goal, which both managers later agreed had been scored from an offside position.
Arsenal trailed 1-0 at the interval, and Moyes claimed Fabregas then confronted referee Lee Mason.
"In doing that, he also questioned the integrity of the referees," claimed the Toffees boss.
"Arsene seems to think that I haven't seen or heard it, but you have to decide who you believe - David Moyes or Arsene Wenger - and I'm happy for people to make their own mind up on that."
Arsenal went on to win 2-1 thanks to second-half goals from Andrey Arshavin and Laurent Koscielny.
Speaking after the match Moyes did not repeat what he claimed to have heard Fabregas saying at half-time but said: "they were disappointing comments from someone who is such a talented footballer".
"I won't go into what they were, but they were deserving of a sending off, 100%.
"If you had said it on the pitch, you should have been off like that, so what is the difference when you are coming down the tunnel?"
The FA said on Wednesday that Fabregas's actions would not be investigated.
Source: BBC Sport
Source: BBC Sport