Secure America's Medicine Act of 2020 This bill requires the Department of Health and Human Services (HHS) to develop and annually update, in collaboration with the Department of Homeland Security and others, a list of critical medications that the federal government should ensure are available in the event of a public health emergency. HHS must develop an initial list and report to Congress on its rationale for including or excluding certain medications. For annual updates, HHS must solicit public input on the contents of the list by consulting with relevant stakeholders and through the Federal Register. In addition, HHS must evaluate the factors that impact the supply of each medication on the list and identify any that may not be available in sufficient quantities in the event of a public health emergency. For any medications with insufficient availability, HHS must take certain actions to include them in the Strategic National Stockpile, and the bill makes changes to the administration of the stockpile to allow HHS to enter into additional types of contracts to acquire these medicines. The bill also requires the Public Health Emergency Medical Countermeasures Enterprise (which coordinates federal preparedness efforts with respect to medical countermeasures) and the Government Accountability Office to address these contracts in their activities and reports related to the stockpile. HHS must also report to Congress on barriers to, and make recommendations to promote, domestic production of critical medications.