Public Health Violence Prevention Act This bill amends the Public Health Service Act to establish the National Center for Violence Prevention in the Centers for Disease Control and Prevention to take over the duties and authorities relating to violence prevention from the National Center for Injury Prevention and Control. The National Center for Violence Prevention, to address violence and prevention as health issues, must establish: (1) the Public Health Violence Prevention Program to award competitive grants to public health departments, schools of public health, criminal justice agencies, and public school districts, with preference given to projects serving areas with the highest rates of violence or the highest increases in violence; and (2) the Private Health Care Sector Violence Prevention Program to award competitive grants to health care providers, community-based organizations, and academic medical centers. The bill establishes the National Health Violence Prevention Program with the following offices and programs: the Office of Applied Violence Prevention Research, the National Advisory Board on Violence Prevention, the Violence Prevention Training Program, the Office of Data Collection and Evaluation, and the Office of Violence Prevention Education and Outreach. The Department of Health and Human Services must rename the Office of Noncommunicable Diseases, Injury, and Environmental Health as the Office of Noncommunicable Diseases, Violence, Injury, and Environmental Health.