The Bayern Munich player suffered a thigh injury in Saturday's friendly against Hungary in Amsterdam and had been hoping to join his team-mates in South Africa this weekend.
"Since the Champions League final (against Inter Milan on 22 May) Arjen hasn't been ready to play a lot or train a lot. It would be wrong to believe he would be ready to face Denmark," Dutch coach Bert van Marwijk told reporters.
"His injury is serious and I don't know when he can join us. We will keep an eye on his progress. We don't want to take any risks. The team's objective is to go as far as possible in the tournament to allow Arjen to play with us during the World Cup."
The Netherlands will also face Japan and Cameroon in their Group E campaign.