“South Dakota is not New York City,” Ms. Noem said at a news conference last week.

But many rural doctors, leaders and health experts worry that is exactly where their communities are heading, and that they will have fewer hospital beds, ventilators and nurses to handle the onslaught.

“We’re behind the curve in rural America,” said Sen. Jon Tester, Democrat of Montana, who said his state needs hundreds of thousands of masks, visors and gowns. “If they don’t have the protective equipment and somebody goes down and gets sick, that could close the hospital.”

Rural nurses and doctors, scarce in normal times, are already calling out sick and being quarantined. Clinics are scrambling to find couriers who can speed their coronavirus tests to labs hundreds of miles away. The loss of 120 rural hospitals over the past decade has left many towns defenseless, and more hospitals are closing even as the pandemic spreads.