When President-elect Donald Trump chose Representative Tom Price of Georgia to be his health and human services secretary, the American Medical Association swiftly endorsed the selection of one of its own, an orthopedic surgeon who has championed the role of physicians throughout his legislative career.

Then the larger world of doctors and nurses weighed in on the beliefs and record of Mr. Price, a suburban Atlanta Republican — and the split among caregivers, especially doctors, quickly grew sharp.

“The A.M.A. does not speak for us,” says a petition signed by more than 5,000 doctors.

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