WASHINGTON—No lawmaker played a more central role in deep-sixing the Republicans’ health-care bill than Rep. Mark Meadows, a previously little-known North Carolina congressman and the leader of an influential group of conservative lawmakers known as the House Freedom Caucus.

That same influence is now being criticized by some in his own party, who say the caucus’s successful effort to scuttle the repeal and replacement of the Affordable Care Act means that Obamacare will remain in place, undermining a central pillar of Republican...