Moyes was asked by BBC reporter Vicki Sparks after the goalless draw with Burnley whether the presence of Black Cats owner Ellis Short at the game had put him under increasing pressure.
"No, none at all," the Scot responded, and thinking the interview was over, he added: "You were just getting a wee bit naughty at the end there, so just watch yourself. You still might get a slap, even though you're a woman. Careful the next time you come in."
Moyes admitted in a press conference on Monday that he 'deeply regrets' the remarks and said he had apologised to both Sparks and the BBC, which was "accepted graciously" by both parties.
"It was in the heat of the moment. I deeply regret the comments I made," he told the press.
"That's certainly not the person I am. I've accepted the mistake. I spoke to the BBC reporter, who accepted my apology."
Moyes has drawn criticism from several quarters, and the FA are requesting an explanation for his conduct at the end of the interview.