National Nursing Workforce Center Act of 2023 This bill sets out a pilot program to support state agencies, state boards of nursing, nursing schools, or other eligible entities with establishing or expanding state-based nursing workforce centers that carry out research, planning, and programs to address nursing shortages, nursing education, and other matters affecting the nursing workforce. The bill also expands the authority of the Health Resources and Services Administration (HRSA) to establish health workforce research centers and specifically requires that HRSA establish a center focused on nursing.