England international Young was booked for tumbling over after a challenge by Crystal Palace's Kagisho Dikgacoi in United's 2-0 win at Old Trafford in the Premier League on Saturday.
Later in the first half, the referee awarded the home side a penalty and sent off Dikgacoi after the pair clashed again, although television replays suggested Young had fallen easily under minimal challenge.
The London club's chairman Steve Parish said Young should have been sent off, while Moyes said he had spoken to the winger over the matter and reiterated his long-held view that there should be retrospective action for diving to clamp down on it.
"You can never be sure it won't happen again, but I've had a word with him privately," Moyes told a news conference on the eve of United's Champions League Group A opener at home to Bayer Leverkusen.
"I've said for many years... we should have retrospective video for diving, I think it would help referees no end."
It will come as little comfort to Parish that even Moyes was unhappy with his own player over diving.
"The only player in the incidents that was honest was Kagisho Dikgacoi and he's sent off and banned for the next match," Parish told the BBC on Sunday.
"Ashley Young has a yellow card and three points and we have no points and one less player to pick from for the next game.
"If preventing a goal-scoring opportunity is a straight red then trying to create one by cheating should be a straight red also. It might have cost us a point that might keep us up.
"(We) need to get some momentum behind a straight red for a dive."
United defender Rio Ferdinand said if there were to be changes to the ways in which diving was handled, the measures would have to be implemented across the world.
"For instance, you go into the Champions League and we're being told in the Premier League that you're not meant to dive but you go in the Champions League and you've got other countries that simulate," he told a news conference.
"It's difficult and obviously as a defender you don't want people to just try and con the referee. If there's contact, in today's game the player has got the right to appreciate that he's been contacted and maybe go down.
"You've got to defend a little bit more with your head and think about how you try and combat the forward because they are cute and know how to do this type of stuff," he added.
"It's a difficult thing for the referee to look at so retrospective viewing and punishments may be the way to go."
Young has gained an unenviable reputation as a player prone to falling over in the penalty area and the latest incident has led to calls for him to be banned.
Former Premier League referee Graham Poll, writing in his column in the Daily Mail, accused Young of cheating.
"He should be banned for five games, then he might stop diving to try and con referees," he said.
Former United manager Alex Ferguson said he had "had a word" with Young last year after several high-profile incidents in which the former Aston Villa player had fallen over.
Last season in the Premier League, 34 yellow cards were shown for diving with former Tottenham Hotspur forward Gareth Bale the worst culprit, picking up six of them.
The Football Association can take retrospective action against players they feel were guilty of simulation, even if the referee deemed nothing was wrong at the time.