Moyes has faced criticism after footage of the conversation, filmed on a mobile phone after Sunderland's 0-0 home game against Burnley on March 18, was published by a tabloid newspaper.
The manager told BBC reporter Vicki Sparks that she "might get a slap, even though you're a woman" after asking whether the presence of club owner Ellis Short in the stands had increased the pressure on him.
Moyes, whose side are rooted to the bottom of the Premier League, later apologised to the reporter -- who has made no complaint -- and also expressed his regret at a news conference on Monday.
Sunderland said in a statement that the comments were "not condoned or excused in any way".
It added, however, that "appropriate action" had been taken at the time.
"David recognised this immediately, proactively bringing the matter to the attention of the CEO and apologising to the reporter," it said.
"The club also spoke with both a senior figure at the BBC and the reporter personally, expressing its profound regret over what had occurred."
Sunderland said the matter had been treated "with the utmost seriousness from the outset" and resolved to the satisfaction of both the reporter and the BBC.
"With both the BBC and the reporter agreeing that appropriate action had been taken at the time, the club continues to fully support David in his role as manager of Sunderland AFC," the statement added.
Rosena Allin-Khan, the opposition Labour Party's spokeswoman for sport, called on Monday for a Football Association investigation.
"If you look at the fact that he wouldn't have said that to a male reporter, and I truly believe that, I think the comments and his behaviour and attitude was sexist," she told BBC radio.
The FA says it is "seeking observations" from the club.
(Reporting by Alan Baldwin, editing by Neville Dalton)
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