Veterans Mental Health Accessibility Act - Makes any veteran who served on active duty in a theater of combat operations during World War II, the Korean conflict, the Vietnam era, the Persian Gulf War, Operations Iraqi Freedom or Enduring Freedom, or any other period of war after the Persian Gulf War, or in combat against a hostile force during a period of hostilities, eligible for hospital care, medical services, and nursing home care for any mental illness through the Department of Veterans Affairs (VA), notwithstanding insufficient medical evidence to conclude that the mental illness is attributable to such service.