Rural Health Innovation Act of 2019 This bill establishes two grant programs to increase access to emergency care in remote rural areas where the nearest emergency department is at least 30 minutes away or inaccessible by road. First, the bill requires the Health Resources and Services Administration (HRSA) to provide grants to establish or maintain federally-qualified health centers or rural health clinics located remote rural areas. Eligible health centers or clinics must (1) employ physicians and professional clinical and support staff; (2) have laboratories, x-ray machines, and cardiac monitors on site; and (3) provide walk-in urgent care, and staging for transport to an emergency department. Further, the bill requires HRSA to provide grants to rural health departments that are located in remote rural areas to support the provision of emergency services, triage and transport to emergency departments, primary care services, or other similar services. Eligible rural health departments must (1) have a nursing staff and medical equipment, (2) agree to provide such emergency or similar services, and (3) submit an estimated cost for the first year of staffing and equipment.