WASHINGTON—President-elect Donald Trump has chosen top officials for health-care policy, picking House Budget Committee Chairman Tom Price (R., Ga.) as secretary of the Health and Human Services Department, the sprawling agency that will likely dismantle Democrats’ 2010 health-care overhaul.

Mr. Trump on Tuesday also named Seema Verma, a consultant who helped Vice president elect Mike Pence negotiate a groundbreaking Medicaid deal with the Obama administration, as the head of the Centers for Medicare and Medicaid Services....