The 18-year-old has become an instant sensation after scoring twice in his first match against Danes Midtjylland in the Europa League last Thursday and repeating the trick on his Premier League debut against Arsenal on Sunday.
The extent of the fuss surrounding the Manchester-born forward has been such that Van Gaal, speaking at a news conference on Tuesday, pleaded for the media to leave the level-headed Rashford alone.
"He is a very modest guy so I don't think that shall be a problem," Van Gaal said. "When you have so much attention from the media it can go in another way but I believe he shall keep his feet on the ground.
"I think the media have to leave him in peace. When you are in front of his house and that kind of stuff, I don't think that is beneficial for a boy of 18."
Van Gaal said it was "remarkable" what Rashford had achieved but added he still had to show consistency.
United face Watford in the Premier League on Wednesday and there is no guarantee, with the prospect of Anthony Martial returning up front, that Rashford will be handed another start.
"Martial and (Chris) Smalling and (Matteo) Darmian are coming back, maybe tomorrow. The chance is there that one of these three plays," Van Gaal said.
As for his own comical role in Sunday's 3-2 win, when his dying swan act at the feet of the fourth official to demonstrate a perceived foul earned him hoots of laughter from fans, Van Gaal did not sound in the mood for a repeat.
"It is not a part of my philosophy because I don't think you can influence the game from that spot," he said.
"But I am not a clown, I am Louis van Gaal, I am a human being and a manager. And the players have to think on the pitch, not the manager."
(Reporting by Ian Chadband; Editing by Tony Jimenez)
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