The Dutch FA said in a statement on Sunday that Stuivenberg had quit as coach of its under-21 team in order to move to Old Trafford.
The 43-year-old has been in charge of the under-21s for the past year and also won the European Championship at under-17 level with the Dutch in 2011 and 2012.
"We are thankful to Stuivenberg for his contribution to the development of Dutch talent. He did a fantastic job with the under-17s and we wish him luck in the new direction that he now takes," said FA technical director Jelle Goes.
Stuivenberg has signed a three-year deal at United, the same length of contract as Van Gaal, and will join fellow countrymen Frans Hoek, Marcel Bout and former Wales winger Ryan Giggs among the backroom staff.
Netherlands meet Argentina in the semi-finals of the World Cup on Wednesday.
(Reporting by Mark Gleeson, editing by Tony Jimenez)
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