Van Nistelrooy will step aside after Sunday's game with Leicester City.
He wrote in the United match programme: "The history at this club is incredible and it will always be a huge honour for me to have played a part in that as a player.
"Now, however, I am here as a coach looking towards the future. When I do that, I see only good things for Manchester United. I absolutely believe that this club has all the ingredients to get back to the levels I knew as a player, but it will take a lot of hard work and determination from everyone around the club to get there.
"From Monday, I will be getting behind our new head coach, Ruben Amorim, and I urge everyone who loves the club to do the same. I have worked with this squad every day since the summer and I can tell you that there is huge potential in these players, from the experienced pros down to the lads coming up through the academy. The coaches, club staff and supporters all have a role in helping those players become the best they can be.
"Of course, it starts with players who hold themselves and each other accountable, because playing for Manchester United requires massive character, hard work and dedication every single day. There is no hiding place, especially with today's media environment, but that is the kind of challenge you have to relish. and meet head-on.
"You show courage, you fight and you show your talent for yourself, for that badge on your shirt and, most importantly, for the greatest fans in the world. Let me tell you, whether you're a player or a coach, there is nothing like representing Manchester United Football Club."