Violence Against Women Health Act of 2019 This bill reauthorizes through FY2023 the Department of Health and Human Services program that awards grants to support the health care system's response to domestic violence. The bill also modifies the program, including by requiring that grant funds be used to (1) support training to address domestic violence in families enrolled in early childhood programs, (2) develop training and resources for behavioral health professionals, (3) support health education training related to labor and sex trafficking, and (4) support the coordination of domestic and sexual violence coalitions. Certain training must prioritize programs administered by the Health Resources & Services Administration's Office of Women's Health. The bill also permits grant funds to be used to establish state pilot programs that address substance use disorder in the context of domestic violence.