DEFIANCE, Ohio — James Waltimire, a police officer on unpaid medical leave, has been going to the hospital in this small city twice a week for physical therapy after leg surgery, all of it paid for by Medicaid.

Mr. Waltimire, 54, was able to sign up for the government health insurance program last year because Ohio expanded it to cover more than 700,000 low-income adults under the Affordable Care Act. He voted for President Trump — in part because of Mr. Trump’s support for law enforcement — but is now worried about the Republican plan to effectively end the Medicaid expansion through legislation to repeal the health care law.

“Originally the president said he wasn’t going to do nothing to Medicaid,” Mr. Waltimire said the other day after a rehab session. “Now they say he wants to take $880 billion out of Medicaid. That’s going to affect a lot of people who can’t afford to get insurance.”

As Republicans in Washington grapple with how to meet their promise of undoing the greatest expansion of health care coverage since the Great Society, they are struggling with what may be an irreconcilable problem: bridging the vast gulf between the expectations of blue-collar voters like Mr. Waltimire who propelled Mr. Trump to the presidency, and longstanding party orthodoxy that it is not the federal government’s role to provide benefits to a wide swath of society.