Rural Health Clinic Modernization Act of 2019 This bill makes a series of changes to criteria, requirements, and payments for rural health clinics under Medicare. Among other things, the bill (1) allows rural health clinics that are not directed by physicians to enter into arrangements with physician assistants or nurse practitioners that generally comply with state laws (regarding scope of practice), rather than other specific requirements; (2) allows rural health clinics to serve as the distant site for purposes of telehealth services; and (3) beginning in 2020, increases the maximum payment rate per visit for rural health clinics.