Gambling Addiction Prevention Act of 2019 This bill directs the Department of Defense (DOD) to implement policies and programs on prevention, education, and treatment of gambling disorder. It requires integration of gambling disorder into existing mental health and substance use disorder programs. If DOD provides gambling activities at a morale, welfare, and recreation facility, it must (1) provide educational materials and a list of resources to assist service members in identifying the risks of gambling and seeking help, and (2) implement measures that encourage behavior consistent with measures used by commercial and tribal gaming regulators. Funding of measures under this bill shall come from gambling activity proceeds generated at DOD morale, welfare, and recreation facilities.