Show Esteem and Respect for Veterans by Increasing Care and Equity Act or the SERVICE Act This bill provides eligibility for hospital care, medical services, and nursing home care for any mental illness through the Department of Veterans Affairs to veterans who served on active duty in a theater of combat operations. Specifically, combat veterans are eligible for such care regardless of (1) whether there is insufficient medical evidence to conclude such illness is service-connected, and (2) time limitations based on discharge or release from active military service.