WASHINGTON—Republicans said they plan to put off any new health-care overhaul plan until after the 2020 election, hoping to avoid a replay of last fall’s bruising campaign and setting up a clear contrast with Democrats looking to expand the government’s role in insurance coverage.

President Trump flagged the shift in a series of late-night tweets in which he continued to attack the Affordable Care Act but said any changes to it would have to wait for a Republican majority in both chambers of Congress.

Heading into the campaign season, the GOP is now on track to pursue targeted measures aimed at lowering drug prices and out-of-pocket costs, as well as attacking such ideas as Medicare for All. Democrats are focused on driving down costs and defending the ACA against GOP-led legal attacks, with some in the party floating broader health-care proposals such as a government-run health system.

“Obamacare has been such a catastrophe because it’s far too expensive,” said Mr. Trump at the White House on Tuesday, offering no details of what he is seeking in a replacement or when it would be revealed. He also warned against expanded government coverage that could potentially replace private insurance. “I see what the Democrats are doing, what they’re planning. It’s a disaster.”