The 22-year-old, who sustained the injury during the Demons' match simulation drills last week, will be sidelined for between four and six weeks.
The recovery time frame puts Petty in significant doubt for the season-opener against the Western Bulldogs on March 16.
"It was a pretty severe (cork)," Melbourne football boss Alan Richardson said.
"We gave it a week off to try and let things settle down but unfortunately that didn't happen, so he had to have surgery.
"The surgery went really well. The surgeon's very optimistic that we're going to get the outcome that we're after.
"We're thinking anywhere between four to six weeks before he'd be right to go again, so that's pretty disappointing for him."
Petty stepped into the vacancy left when Adam Tomlinson suffered a season-ending knee injury last year and was an important contributor in Melbourne's drought-breaking premiership.
He signed a three-year contract extension in October.
Petty's injury is one of few setbacks for Melbourne during a strong summer preparation, in an ominous sign for the rest of the competition.
"The rest of the list is really healthy, so we're in a really good spot," Richardson said.
"From a conditioning perspective, with a combination of endurance, speed, strength, just about everyone has improved.
"We've been rapt with the attitude and commitment they've had to their off-season programs and have seen them come back in really great shape."
Meanwhile, Hawthorn forward Luke Breust suffered a suspected concussion during an intra-club match on Thursday morning.
Breust, who was crunched in a marking contest, is likely to miss the Hawks' practice match against Collingwood next week.
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