Protecting American Innovation Act This bill establishes certain limitations on modifying trade agreements, including those related to the TRIPS Agreement (i.e., the Agreement on Trade-Related Aspects of Intellectual Property Rights). The TRIPS Agreement contains obligations for World Trade Organization members to protect patents and other intellectual property rights. Specifically, the bill prohibits the President (or any U.S. official, employee, or agent) from negotiating or concluding any withdrawal, suspension, or modification to a trade agreement that adversely affects the rights of the United States or U.S. persons under a trade agreement with China or Russia. Additionally, the bill states that a TRIPS waiver to address the COVID-19 pandemic shall not take effect if (1) the President does not submit required reports from the Department of Commerce and the Department of Defense (DOD), (2) the Commerce report concludes that the TRIPS waiver will not result in an increase in global vaccine access, or (3) the DOD report concludes that the TRIPS waiver would adversely impact U.S. national security. Prior to entering into a negotiation with a trading partner concerning a suspension of or modification to a trade agreement, the bill requires (1) the U.S. Trade Representative to publish certain information in the Federal Register and allow for public comment, and (2) the President to provide written notice and consult with Congress. Further, the President shall not enter into any suspension of or modification to a trade agreement unless the President has complied with such consultation requirements and receives congressional approval.