Januzaj appeared to be stamped on by Sascha Riether during injury-time in Saturday's 3-1 win at Fulham.
The incident was not spotted by referee Lee Probert, with Moyes claiming immediately afterwards he had not seen it either.
Riether was charged with violent conduct by the Football Association over the incident on Monday, becoming the first player to be charged under a pilot scheme where a panel of three former referees reviews video evidence of incidents not seen by match officials.
Having viewed the replays himself, the United boss feels Januzaj needs more protection than he is presently receiving.
"I had not seen it until after the game," Moyes said.
"The distance we were from the dug-out made it impossible to take it in.
"It is a bad stamp and it does look deliberate."
Speaking before being made aware of the FA charge, Moyes said of the incident: "That is for the referee to deal with on the day as far as I am concerned. It is over now.
"But I do think Adnan is going to need a little bit of protection or we are going to find that he is getting kicked up and down.
"I do think he deserves a little bit more than he is getting at this present time."
Moyes accepts the nature of the modern game means Januzaj will be targeted.
Having already reprimanded Ashley Young for diving this season, it is clear the Scot will not tolerate the teenager play-acting in an attempt to gain an advantage.
"We have spoken to him about it," said Moyes.
"He gets kicked up and down in training every day. But he gets up.
"He has learned from the good professionals that he has got to get on with it and accept it.
"He only needs to look at Lionel Messi, who is probably the best in the world at getting kicked, getting up and getting on with it and continually producing magic all the time.
"I am not in any way comparing him to Lionel Messi but I am saying we hope the way he gets up after he has been kicked is the way Adnan continues to behave."
Moyes did confuse the issue slightly when he was asked to clarify his comments.
"It is not a case of asking for protection," he said.
"We are just saying that somewhere along the line, people will need to let the boy, if he is going to keep playing in the fashion and style we want, he is going to have to get some protection from it.
"But we will never ask for any.
"It is up to the referees to do what they think is right."