Franchitti was pulled out of his car, put on a stretcher and taken to Memorial Hermann Texas Medical Center in Houston after his vehicle sailed off the street circuit-style track and into a catch fence on the final lap.
The wreck also involved drivers E.J. Viso and Takuma Sato.
The 40-year-old Scotsman was later diagnosed with a spinal fracture, a fractured right ankle and a concussion.
The Houston Fire Department told CNN two of the 13 injured spectators were sent to local hospitals with minor injuries. The rest were treated at the scene.
An IndyCar timing and scoring official also suffered minor injuries in the incident.
Will Power won the second of two weekend races at Houston's Reliant Park Sunday.