Romo said he will undergo tests to determine the damage, but Cowboys owner and general manager Jerry Jones said the quarterback could miss six to eight weeks.
"We'll see if surgery is required," Jones said. "If not, he's going to have some time out. [Troy] Aikman was out four weeks with this and our [trainers] initially think six [weeks], possibly as long as eight [weeks], but we'll see what his examination is."
Romo went down in the second quarter on a hit by Giants' linebacker Michael Boley. The Cowboys, who were leading 10-7 at the time, are now 1-5.
Veteran quarterback Jon Kitna replaced Romo, and will be the starter while Romo heals. The Giants are now 5-2 and in first place in the NFC East.