Securing America's Medicine Cabinet Act of 2020 This bill encourages the development and approval of advanced pharmaceutical manufacturing technologies and designates certain university research centers to assist in developing such technologies. The Food and Drug Administration (FDA) must continue to evaluate and approve new drug manufacturing technologies included in an application for drug approval and expedite the development and implementation of such technologies. The FDA must designate such a technology as an advanced manufacturing technology if it is likely to (1) prevent or resolve a drug shortage, (2) maintain an adequate supply of critical medications for national emergencies, or (3) promote the adoption of innovative approaches to drug design and manufacturing. The sponsor of such a designated technology must provide the FDA with certain related scientific evidence. After receiving this evidence, if the FDA validates the technology for a proposed use, then the sponsor may use the validated technology across multiple manufacturing product lines within the same use context without obtaining additional FDA validation. The FDA must designate certain institutions of higher education as National Centers of Excellence in Advanced Pharmaceutical Manufacturing. Among other requirements, such centers must demonstrate the ability to provide federal agencies with technical assistance and to train a future workforce in such technologies.