The 53-year-old Scot issued a public apology to BBC Newcastle and Radio Five Live reporter Vicki Sparks on Monday after video footage emerged of a controversial exchange between the pair following a post-match interview.
On Tuesday afternoon, the Black Cats issued a statement confirming their position on the matter and offering their continued backing to Moyes.
It read: "Sunderland AFC would like to reiterate the actions taken following the Burnley game on March 18.
"The exchange between the manager and a BBC reporter was wholly unacceptable and such actions are not condoned or excused in any way.
"David recognised this immediately, proactively bringing the matter to the attention of the CEO (Martin Bain) 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.
Read the full article on eurosport.co.uk: Sunderland stand by Moyes but call 'slap' remark 'wholly unacceptable'