Mobile Health Record Act of 2019 This bill requires the Centers for Medicare & Medicaid Services (CMS) to establish a program that enables Medicare enrollees to connect claims data with trusted applications, services, and research programs. The program must allow an enrollee to access claim information through a mobile health record application that is chosen by the enrollee and that is also approved by the CMS, in accordance with specified requirements. Information must be made available to such applications in a similar manner as under the current Blue Button 2.0 program. The bill also allows states that choose to develop or purchase similar applications for Medicaid enrollees to receive federal payment for the associated costs. Currently, the CMS administers the Blue Button program, which allows Medicare enrollees to download and save health information on personal computers or other devices. The CMS subsequently launched the Blue Button 2.0 program, which allows enrollees to link health information with mobile applications (these applications are posted on the CMS website, but are not endorsed by the CMS).