The 31-year-old Manchester United player has won 94 caps for the country and played in the 2014 World Cup final defeat by Germany.
Chelsea’s Willy Caballero, with two caps, and River Plate’s Franco Armani, who is uncapped, are Argentina’s reserve goalkeepers.
Tigres’ Nahuel Guzman has been called into the squad as Romero’s replacement.
Argentina’s Football Association said Romero will have surgery on his injured right knee.
Argentina begin their World Cup campaign against Iceland on 16 June before games against Croatia and Nigeria in Group D.