The Romanian's life is not in danger but he will have to stay under observation in a Verona hospital for some time after an operation to relieve a haematoma and take out bone fragments.
"Everything went normally and therefore we are quite confident the patient can recover without problems," doctor Sergio Turazzi said in a statement.
The loss of the dependable Chivu is a blow to the Serie A leaders, who have no like-for-like replacement. Inter's right-footed fullback Davide Santon is also injured and Javier Zanetti is needed in midfield.
"We hope (Chivu) can come back as soon as possible but I think he will be out of action for a while," Inter coach Jose Mourinho told the club's TV channel.
Media reports said Inter were chasing Lazio left back Aleksandar Kolarov even before Chivu's injury but Lazio president Claudio Lotito has said the Serb is not for sale.
Mourinho, who has already brought in Macedonia forward Goran Pandev, also hopes the club can recruit another midfielder in the January transfer window with the little-used Patrick Vieira on the verge of leaving.
Manchester City have been strongly linked with the Frenchman in the media.
