What troubles many Republican strategists is the specter of the party’s most reliable voters being bombarded by reminders of their leaders’ failure to address the health law. They fear a recurring story line sure to pop up every time insurance premiums increase, providers leave local networks, or, most worrisome, Republicans fund President Barack Obama’s signature achievement.

Conservatives, many of whom opposed the House repeal bill, now warn that it is untenable to stand pat on the issue — and that lawmakers will face retribution if they do not return to the repeal-and-replace effort.

“If people are looking at a situation where there’s no action on this, there are going to be conversations about primaries,” warned Michael A. Needham, the chief executive of Heritage Action for America, the Heritage Foundation’s political arm, which worked to scuttle the Republican health bill last week.

That Republicans even find themselves in such a quandary just over two months after Mr. Trump was sworn in is at once extraordinary and not altogether surprising. Republicans who were then in office opposed the Affordable Care Act when it was enacted in 2010, yet they were paralyzed in efforts to undo it.