Reinforcing Enhanced Support through Promoting Equity for Caregivers Act of 2023 or the RESPECT Act of 2023 This bill modifies the caregiver programs of the Department of Veterans Affairs (VA) and addresses the provision of support for caregivers participating in such programs. The bill expands (1) the Program of Comprehensive Assistance for Family Caregivers to include services related to neurological disorders and additional mental health services, and (2) the Program of General Caregiver Support Services to include services related to neurological disorders. The VA must establish a process and requirements for its clinicians to document incidents in which veterans in the family caregiver program present at a facility for treatment for an emergent or urgent mental health crisis or are assessed to be at risk for suicide. The bill sets forth the VA's process and requirements for determining or reassessing the eligibility of a veteran and caregiver for the family caregiver program, including with regard to the provision of documentation and evidence related to mental health, psychological trauma, or a neurological disorder. The VA must waive the reassessment requirement under specified circumstances, such as when a veteran's serious injury is caused by a degenerative or chronic condition. The VA is authorized to award grants to implement, coordinate, improve, or enhance mental health counseling, treatment, or support for family caregivers of veterans participating in the family caregiver program. The Government Accountability Office must report on the provision of mental health support to caregivers of veterans.