Supporting Disabled National Guardsmen Act This bill provides disability pay and medical care for members of the National Guard who were physically disabled as a result of state active duty. State active duty includes training or other duty in service to the governor of a state for which the member is not entitled to federal pay, but excludes required drills and field exercises. The bill expands eligibility for Department of Defense retired pay to such members who are physically disabled as a result of state active duty after September 23, 1996. Such pay must be reduced if it is determined to duplicate any other federal or state benefit to such members based on disability. The bill also makes such members eligible for hospital care and medical services from the Department of Veterans Affairs (VA) to treat the disability and any illness or condition arising from the disability. The VA may provide reimbursement for hospital care or medical services provided to such members only after a member or the provider of care has exhausted (without success) all claims and remedies reasonably available against a third party.