Gakpo, who has scored 25 times in 56 appearances for the Dutch, hurt his ankle in Sunday’s 2-1 win over Serbia in Belgrade, going off after 17 minutes of the Nations League clash following a strong challenge by Sasa Lukic.
The KNVB said he would return to his club Liverpool and new coach Xavi Hernandez would not call up a replacement.
Gakpo netted the late equaliser in last week’s 1-1 Nations League draw with Germany, which was the first game since the World Cup for the Nethelrands and a debut for Xavi.
The Dutch play Greece in Thessaloniki on Thursday followed by a home clash against Serbia in Eindhoven on Sunday.
Gakpo misses a return to the Philips Stadion where he played for PSV Eindhoven before moving to the Premier League.