Veterans' Culturally Competent Care Act This bill requires the Department of Veterans Affairs (VA) to establish standards and requirements for the provision of mental health care by non-VA providers in the Veterans Community Care Program (VCCP). The established standards must be the same as the standards applicable to VA employees who provide mental health care. The VA must require non-VA mental health care providers to complete training courses on military culture, core competencies for health care professionals, suicide evaluation and management, post-traumatic stress disorder, traumatic brain injury, and military sexual trauma after the provider becomes approved under the VCCP. The VA must also (1) identify additional training areas for which to develop courses, and (2) require the non-VA providers to complete such courses.