Medical Student Education Authorization Act of 2022 This bill establishes matching grants awarded by the Health Resources and Services Administration (HRSA) to expand and support medical education with a particular focus on training students to serve as primary care physicians in tribal, rural, or medically underserved communities. HRSA must award the grants to public institutions of higher education located in states that rank in the top quartile of states based on projected unmet demand for primary care providers. In awarding the grants, HRSA must give priority to recipients that (1) are located in states with two or more federally recognized Indian tribes, and (2) demonstrate a public-private partnership. Recipients must use grant funds for specified activities that include developing curricula, expanding partnerships with health care providers and community-based organizations, and providing scholarships.