MINNEAPOLIS—Republicans have often won support in recent elections by promising to repeal the Affordable Care Act. This year, Democrats hope to turn the tables by pushing the opposite goal—not just keeping the health law, but expanding government’s role in health care.

The tactic, which carries political risk as well as opportunity, is playing out in places such as Minnesota, a state won narrowly by Hillary Clinton in 2016 that is facing a governor’s race, two Senate contests and five close House races. Democrats need to gain...