The Liverpool centre-back completed 90 minutes on Saturday as Ronald Koeman's side battled to a 0-0 draw in Northern Ireland, securing their place at Euro 2020 with a game to spare in qualification.
Top spot in Group C is still up for grabs heading into Tuesday's game versus Estonia at the Johan Cruijff ArenA, with Germany two points ahead going into their own match against Northern Ireland, although captain Van Dijk will not take part.
"National team coach Ronald Koeman cannot rely on VirgiI van Dijk on Tuesday against Estonia," read a short statement from the Royal Dutch Football Association (KNVB).
"The captain of the Dutch national team has to [miss] the game due to personal circumstances and has since left the Oranje training camp."
Van Dijk's next engagement with Premier League leaders Liverpool is a trip to Crystal Palace on Saturday.