The 43-year-old recently worked under Van Gaal as Netherlands U21 coach, having previously won the European Championships with the Dutch U17 team in 2011 and 2013.
Stuivenberg, who is also former Feyenoord academy manager, will be Van Gaal's assistant coach at United.
Van Gaal has already appointed Frans Hoek and Marcel Bout to his staff, and will start work at Old Trafford after the World Cup is over.
Netherlands made it through to the semi-finals of the tournament on Saturday night when they beat Costa Rica on penalties.
And Van Gaal’s team will now play Argentina on Wednesday in Sao Paulo with Brazil or Germany to come after that in either the final or the third-place play-off.