Reproductive Health Care Training Act of 2023 This bill establishes a grant program that supports training for providers of abortion care. Specifically, the Health Resources and Services Administration must award grants to health professional schools, academic health centers, or other nonprofit health care safety net providers to support training in abortion care, if such training is authorized in the state in which they are located. Funds may be used to train students, residents, and advanced practice clinicians in abortion care and to connect them to appropriate entities to provide such care, among other activities.