LGBTQI+ and Women's History Education Act of 2023 This bill directs the National Museum of American History of the Smithsonian Institution (NMAH) to carry out activities to support LGBTQI+ (sexual and gender minority populations) history and women's history education programs. Specifically, the NMAH must, among other things, carry out the following activities: the development and national dissemination of accurate, relevant, and accessible national resources for classrooms to administer LGBTQI+ and women's history education in an inclusive and intersectional approach; the development, national dissemination, and implementation of principles of sound pedagogy for administering LGBTQI+ and women's history education; the development, operation, and expansion of a teacher fellowship program to cultivate and support leaders in LGBTQI+ and women's history education; and engagement with state and local education leaders to encourage the adoption of resources supported under this bill into curricula across diverse disciplines. In addition to the activities described in this bill, the museum must develop and maintain on its website content designated for educators, students, and families to improve their awareness and understanding of LGBTQI+ and women's history education.