Leonard, who underwent a CAT scan Thursday morning, suffered the injury during the first half of the Spurs' loss to the Oklahoma City Thunder on Wednesday night.
Spurs teammate Danny Green has a similar injury and is also expected to miss a month.
"It's tough, we're dropping like flies right now,'' Spurs center Tim Duncan said. "Losing Kawhi was huge, especially in a game like tonight.''
After remaining relatively healthy in reaching the NBA Finals last season, the Spurs have been plagued by a rash of injuries in the past month.
Leonard joins starting center Tiago Splitter and former starter Green on the injured list. Tony Parker still has a bruised shin that limits his mobility, and long-range shooting forward Matt Bonner has to wear a mask after breaking his nose Friday against the Portland Trail Blazers.
"[Losing Leonard] is a big loss for us,'' Parker said. "We are already down Tiago and Danny, but everyone goes through injuries and we have to deal with it. Other guys are going to have to step up, but we just have to keep playing.''