U.S. coach Bob Bradley said the defender would be out for up to four months after tearing a patellar tendon in his left knee.
"I'm very angry because once again the national teams take our players and we have to pay the consequences of injuries," Milan chief executive Adriano Galliani was quoted as saying by Italian media on Thursday.
"We demand compensation from the American Federation because we cannot pay the salary of a player who cannot play for six months."
The player, who has hardly featured at Milan since joining on a free transfer from Standard Liege in the close season, should recover in time for next year's World Cup finals.
"We've had two days of tough news," said Bradley.
"'Guch' is such an important part of our team. He's young, he's healthy. Our doctors are good but nonetheless, it's another setback for us," he said.
