Servicemembers Mental Health Care Commission Act - Establishes a commission on veterans and members of the Armed Forces (members) with post-traumatic stress disorder (PTSD), traumatic brain injury (TBI), or other mental health disorders caused by military service. Requires the commission to: (1) oversee the monitoring and treatment of veterans and members with such disorders; and (2) conduct a study of the long-term adverse consequences of such disorders for such veterans and members.