Save America's Rural Hospitals Act This bill establishes and otherwise modifies payment requirements for rural health care providers under Medicare and Medicaid. For example, the bill allows additional hospitals to qualify as critical access hospitals (CAHs) that receive special payment under Medicare. A hospital may qualify if it is a small, rural hospital that (1) serves a health professional shortage area, or a high number of low-income individuals or Medicare or Medicaid beneficiaries; (2) has experienced financial losses for two consecutive years; and (3) attests to having a strategic plan to address financial solvency. In addition, the Center for Medicare and Medicaid Innovation must test a new delivery and payment model for rural hospitals that promotes financially sustainable access to care and must establish a corresponding transition process for CAHs. The bill also permanently (1) increases payments for Medicare-dependent hospitals and low-volume hospitals, (2) increases payments for Medicaid primary care services in rural areas, and (3) removes the 96-hour physician certification requirement for inpatient CAH services under Medicare. It also exempts Medicare payments to rural hospitals from sequestration.