WASHINGTON—The Supreme Court agreed to once again decide the fate of the Affordable Care Act, but it likely won’t do so before the 2020 presidential election.

The court on Monday announced it would consider an appeal by a group of Democratic-led states that are defending the 2010 health-care overhaul legislation, a signature achievement for President Obama. The ACA is under legal attack from a group of Republican states, which found a new tactic for challenging the law in 2018—based on a provision requiring Americans to buy...