Media playback is not supported on this device
After a draw with Burnley on 18 March, Moyes was asked by Vicki Sparks if the presence of owner Ellis Short put extra pressure on him.
He said "no" but, after the interview, added Sparks "might get a slap even though you're a woman" and told her to be "careful" next time she visited.
"It was in the heat of the moment," said Moyes.
Both Moyes and Sparks were laughing during the exchange and the former Everton and Manchester United manager later apologised to Sparks, who did not make a complaint.
Speaking in a news conference on Monday, he added: "I deeply regret the comments I made.
"That's certainly not the person I am. I've accepted the mistake. I spoke to the BBC reporter, who accepted my apology."
The BBC confirmed that Moyes and Sparks had spoken about the exchange and the issue had been resolved.
A spokesman said: "Mr Moyes has apologised to our reporter and she has accepted his apology."
However, shadow sports minister Dr Rosena Allin-Khan called on the Football Association to get involved.
"This is disgraceful. David Moyes cannot get away with these sexist threats - the FA must take action immediately," she said.
Former England striker and BBC Match of the Day presenter Gary Lineker also condemned Moyes' behaviour.
"Moyes incident highlights a tendency for some managers to treat interviewers with utter disdain. Pressured job. Well rewarded. Inexcusable," he said.
A statement from Women in Football said it was "deeply disappointed and concerned" but "pleased that David Moyes has apologised".
It added: "No-one should be made to feel threatened in the workplace for simply doing their job.
"We hope that the football authorities will work with us to educate football managers and those working within the game to prevent this kind of behaviour."
Sunderland are bottom of the Premier League on 20 points, eight points from safety, going into a game at Leicester City.
Sunderland boss David Moyes says he "deeply regrets" telling a BBC reporter she might "get a slap".