WASHINGTON—A Republican push to pass a sweeping health-care law experienced another setback as Senate leaders said they would delay a vote set for this week, sparking fresh doubts about whether congressional leaders can muster support for a marquee GOP policy priority.

President Donald Trump and party leaders in Congress were hoping the Senate would vote this week on a plan to overturn parts of the 2010 Affordable Care Act and make other changes to the health system. But Senate leaders announced a delay after Sen. John McCain...