The England international was pictured partying in the Northern Ireland capital days before their FA Cup victory over Newport County.
Ten Hag claimed he missed training due to illness but it was later reported that Rashford was drinking for 12 hours and passed out in a hotel bed at 3am.
United released a statement saying Rashford has 'taken responsibility' and the matter has now been closed, with the player back in training.
Ten Hag will welcome him back to the side to face Wolves and did not want to comment on the situation much further in a tense Manchester United press conference on Wednesday.
"He has taken responsibility and for the rest it's an internal matter. Case closed," he told the media.
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When asked about Rashford again, he insisted: "As I said, case closed."
Ten Hag was also questioned on discipline within his squad but hinted that the winger was not lacking respect: "It's nothing to do with that but as I said, let's focus on the game, let's move on.
"The statement is clear and we have to focus on winning football games."
He added: "Nothing to do with [discipline] with me but in football, you need discipline and that is on the pitch but also off the pitch.
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"There is a line between, when every player, every professional knows this, what is required. We have to focus on winning football games and that's well that matters and every top professional knows what is required."
Ten Hag previously disciplined Jadon Sancho by banishing him from the first-team after the player hit out at his boss on social media.
Manchester United manager Erik ten Hag gives numerous blunt responses to questions about Marcus Rashford's reported 'illness'
He has since made a return to Borussia Dortmund, signing for his former club on loan until the end of the season.
Rashford could be set to appear in the United starting XI on Thursday night for their trip to Wolves.
The 26-year-old has scored four goals this season, a tally that is far less than the 30 he netted in an excellent 2022/23 campaign.