COVID-19 International Response and Recovery Act of 2020 This bill addresses the role of the United States in the global response to COVID-19 (i.e., coronavirus disease 2019) and future pandemics. Specifically, the bill authorizes measures to facilitate the repatriation of U.S. nationals who are abroad during the COVID-19 pandemic. The bill also provides support for international organizations, including through (1) providing payments to organizations that address global humanitarian needs, and (2) advocating for the establishment of a World Bank Trust Fund to aid countries responding to epidemics and pandemics. Further, the bill establishes measures related to refugee, humanitarian, and migration matters during the pandemic, such as those facilitating humanitarian assistance to countries affected by COVID-19, addressing threats to women and girls, and restricting deportation of foreign nationals. The bill establishes leadership roles for the U.S. Agency for International Development, the State Department, and the Centers for Disease Control and Prevention should an infectious disease with pandemic potential break out outside the United States. The bill also creates the role of Special Advisor to the President for Global Health Security, who shall coordinate the U.S. response to global health security emergencies. Further, the bill requires the Department of Homeland Security to automatically extend certain immigration-related statuses, such as employment authorizations, that would otherwise expire during the COVID-19 pandemic. The bill also allows a Peace Corps volunteer whose period of service was terminated due to COVID-19 to file for pandemic-related unemployment benefits.