Zanetti underwent tests at Pavia's Policlinico San Matteo after being forced off the pitch in the 17th minute at the Stadio Renzo Barbera.
The 39-year-old Argentinian looks set to face a lengthy absence, but insisted he had no plans to call it a day.
"My goal is to come back stronger than before, and I believe," he said on Inter's website. "I had to change tyres after so many miles... I'm sorry for how the season has gone.
"Injuries can happen and they're part of this profession. The most important thing is the team. We'll decide when surgery will take place.
"They still haven't given me the timescale involved, but my career certainly isn't over.
"What matters is healing well. What I want most is to be back with my team-mates."
Sunday's result dealt a blow to the Nerazzurri's hopes of securing European football next season, leaving them in seventh spot - two points off a Europa League berth.
Josip Ilicic scored the only goal of the game after 10 minutes to lift the hosts out of the bottom three on goal difference.
Zanetti continued: "I'm sorry I can't give (my team-mates) a hand in the final part of the season, but I'm confident they'll be able to qualify for the Europa League.
"I appreciate the messages from the fans, as well as those from president (Massimo) Moratti, my team-mates and the many others who have shown their support."