The 27-year-old sustained the injury while training for this week's Monte Carlo Masters.
Despite being ruled out of the event at Roland Garros, Fleming is hoping that he can return in time to partner Ross Hutchins at Wimbledon in June.
"Bad news folks. The injury is a stress fracture in my left tibia bone. I've withdrawn from the tournament here with an injury," Fleming wrote on Twitter.
"Going to do all I can to minimise the time I'm out injured. Focused on rehab now and hoping to be back on court in 6-8 weeks. Aiming for 6 weeks not 8."
Fleming and Hutchins are currently ranked number ten in the world standings.