Sailor Standard of Care Act of 2023 This bill requires the Department of the Navy (DON) to implement policies and procedures related to mental health and quality of life for members of the Navy. Specifically, the DON must provide additional resources to any unit with greater than 15 members designated for limited duty at a given time. (Generally, limited duty status is designated for medical reasons and can be permanent or temporary.) The DON must require that members who are designated for limited duty receive a mental health screening upon such designation and once every 60 days for the duration of the designation. The DON must establish a framework to ensure (1) centralized leadership for the management of members of the Navy who are designated for limited duty, and (2) members who have suicidal ideations or other mental health care needs have smooth transitions during transfers between units or while designated for limited duty. The DON must brief Congress on the average number of days for the medical separation of a member previously designated for limited duty and recommendations for steps the DON may take to streamline and expedite timely separation. The Office of the Assistant Secretary of the Navy for Manpower and Reserve Affairs must establish and maintain a dashboard to track quality of life programs (e.g., childcare programs) of the DON and the rate of usage of each program. The DON must study recent cases in which multiple suicides occurred within 30 days of each other among members assigned to the same unit or command.