LGBTQI+ and Women's History Education Act of 2022 This bill directs the National Museum of American History of the Smithsonian Institution to carry out, among other things (1) 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; (2) the development, national dissemination, and implementation of principles of sound pedagogy for administering LGBTQI+ and women's history education; and (3) the development, operation, and expansion of a teacher fellowship program to cultivate and support leaders in LGBTQI+ and women's history education. In addition to the activities described in this bill, the museum shall 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.