Moyes has expressed his regret and apologised to BBC reporter Vicki Sparks after saying "you might get a slap, even though you're a woman".
"The exchange between the manager and a BBC reporter was wholly unacceptable and such actions are not condoned or excused in any way," the statement read.
"David recognised this immediately, proactively bringing the matter to the attention of the CEO and apologising to the reporter. The club also spoke with both a senior figure at the BBC and the reporter personally, expressing its profound regret over what had occurred.
"The matter was treated with the utmost seriousness from the outset and the swift and decisive action taken by the club and the manager at the time ensured that it was resolved to the satisfaction of the reporter and the BBC, which was the priority.
"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."