The experienced defender could be out for up to three months with the injury, which was sustained as the Serie A side opened their International Champions Cup campaign with a 2-1 win at the Estadio Mestalla.
A Milan statement revealed that the 32-year-old will miss "70-90 days, barring complications".
His absence leaves the club with just two out-and-out centre-backs in their squad in the shape of Philippe Mexes and Cristian Zapata.
The Serie A outfit also announced that Robinho could be out for two weeks after suffering a thigh strain in the same game.
The Brazilian was forced off after just 27 minutes and is now a major doubt for Milan's UEFA Champions League play-off, the first leg of which will take place on August 20.
Robinho had looked set to leave Milan ahead of the new season, and re-join boyhood club Santos, but he opted to sign a three-year deal at San Siro after the move fell through.
