Maybe. If the system is set up like Medicare, it would cover many prescription drugs. Lower prices are possible if the state negotiates them or regulates them in some other way.
Drug makers have generally opposed universal health care. In 2016, the Pharmaceuticals Research and Manufacturers of America trade group spent $64 million in California fighting a ballot proposal that would have required state agencies to pay the same for drugs as the Department of Veterans Affairs, which is considered a single-payer system. Voters said no.
The question about medicine prices looms large. The U.S. spends more per capita on prescription drugs than other high-income countries—even though usage is about the same in all the countries.