Climate Change Health Protection and Promotion Act of 2021 This bill requires the Department of Health and Human Services (HHS) to take a series of actions to address the potential impacts of climate change on public health. Specifically, HHS must develop a strategic action plan and a climate and health program to ensure the public health and health care systems in the United States and other nations are prepared for, and can respond to, the impacts of climate change on health. HHS must revise the plan periodically to reflect new scientific developments and information. Additionally, HHS must establish a permanent science advisory board to provide (1) scientific and technical recommendations on the impacts of climate change on public health, and (2) advice for purposes of issuing the plan and conducting the program.