Strengthening the National Disaster Medical System Act of 2019 This bill makes several changes to the National Disaster Medical System, which is administered by the Department of Health and Human Services Office of the Assistant Secretary for Preparedness and Response. Among other changes, the bill requires an agency review of the capacity of the system's workforce of intermittent disaster-response personnel to meet the needs of public health emergencies, and temporarily alters the appointment process to address workforce shortages.