Protecting Service Members and Military Families' Access to Reproductive Care Act of 2023 This bill provides statutory authority for a Department of Defense (DOD) policy that provides for the granting of an administrative absence at the request of a member of the Armed Forces for purposes of receiving non-covered reproductive health care or to accompany a spouse or other dependent who receives non-covered reproductive health care. Under the bill, non-covered reproductive health care means care not authorized to be performed at a DOD facility, including an abortion or assisted reproductive technology (e.g., in vitro fertilization). An administrative absence granted for such purposes may not exceed 21 days. DOD is authorized to prescribe regulations under this bill, which must provide that (1) no additional requirements may be imposed on a member requesting an administrative absence for non-covered reproductive health care, and (2) DOD and the approval authority of a member must prioritize the privacy of the member (or dependent). The bill also authorizes DOD to provide an allowance for travel expenses in connection with a member's or dependent's travel to receive non-covered reproductive health care.