Protect Sexual and Reproductive Health Act of 2022 This bill requires the Department of Health and Human Services (HHS) to undertake activities to promote access to sexual and reproductive health and well-being. The bill renames HHS's Office of Population Affairs as the Office of Sexual and Reproductive Health and modifies its responsibilities. Among other activities, the office must develop and implement a strategy to promote sexual and reproductive health and well-being. This strategy must include recommendations to integrate sexual and reproductive health equity and reproductive justice into federal programs. HHS must also award grants for improving access to sexual and reproductive health care to nonprofit or community-based organizations that assist individuals seeking abortion services through programs that are unbiased and medically and factually accurate. Grant funds may not be used to pay for abortion services. Additionally, HHS and the Director of the White House Gender Policy Council must jointly establish an interagency task force to coordinate and promote federal programs and activities related to sexual and reproductive health and well-being.