Stopping The Outrageous Practice of Surprise Medical Bills Act of 2019 or the STOP Surprise Medical Bills Act of 2019 This bill prohibits health care providers and health insurance plans from billing enrollees in excess of the in-network amount for specified health care services provided out-of-network. Specifically, a plan or provider may not charge an enrollee more than the in-network amount for services that are emergency services provided by an out-of-network provider or at an out-of-network facility; nonemergency services provided at an in-network facility by an out-of-network provider; or nonemergency services provided out-of-network to an enrollee who initially enters through an emergency room for emergency services, except under specified circumstances. Additionally, health insurance plans must pay the median in-network amount, minus the enrollee's in-network cost-sharing amount, directly to a health care provider. The bill also establishes an independent review process to resolve billing disputes between insurance plans and providers.