Save Rural Hospitals Act This bill amends titles XVIII (Medicare) of the Social Security Act (SSAct) to increase payments to, and modify various requirements regarding, rural health care providers under the Medicare program. Among other provisions, the bill: (1) reverses cuts to reimbursement of bad debt for critical access hospitals (CAHs) and rural hospitals, as well as alters certain requirements with regard to CAHs; (2) extends payment levels for low-volume hospitals and Medicare-dependent hospitals (MDHs); (3) reinstates revised diagnosis-related group payments to MDHs and sole community hospitals (SCHs), as well as reinstates hold harmless treatment for hospital outpatient services for SCHs; (4) delays the application of penalties for a rural hospital's failure to become a meaningful electronic health record user; (5) makes permanent increased Medicare payments for ground ambulance services in rural areas; (6) alters certain supervision requirements for therapeutic hospital outpatient services; (7) modifies requirements related to the use and payment of recovery audit contractors; and (8) establishes a program under which rural hospitals meeting specified requirements may be eligible for enhanced payment for qualified outpatient services. In addition, the bill amends the Balanced Budget and Emergency Deficit Control Act of 1985 to eliminate Medicare sequestration for rural hospitals. The bill also amends title XIX (Medicaid) of SSAct to extend Medicaid primary care payments. With respect to both the Medicare and Medicaid programs, the bill eliminates disproportionate share hospital payment reductions for rural hospitals. The bill also amends the Public Health Service Act to authorize several competitive grant programs to assist eligible rural hospitals.