The Bayern Munich forward sustained the injury in the 6-1 warm-up win over Hungary on the weekend and will now miss the clash with the Danes in Johannesburg on Monday.
“Since the final of the Champions League, Arjen has hardly played,” said Dutch coach Bert van Marwijk.
“It is illusory to believe that he will be ready to face Denmark.
“Whatever one may say, his injury is serious and I do not know when he can join us.
“The team’s goal now is to go as far as possible in the competition to allow Arjen to play with us during this World Cup.”