For Dr. King of UCSF, who was watching on C-SPAN, the return to the White House was an opportunity to observe how the president breathed.

“As a pulmonologist, he did two things for me: He did a walk test, and he did a stair-climbing test,” Dr. King said, adding that even with the availability of sophisticated technology, lung doctors still rely on these old-fashioned tests “to just get a picture of how the patient’s doing.”

Dr. King said what he saw concerned him. Mr. Trump paused twice while walking across the lawn — whether to wave to cameras or to catch his breath, he said was not clear — and then appeared to be gasping for breath at the top of the stairs. He and others said Mr. Trump used his neck muscles to help him breathe, a classic sign that someone’s lungs are not taking in enough oxygen.

“This suggests that something’s going on, I don’t know what or to what extent,” Dr. King said. “Any pulmonologist, I think looking at that would give them pause. And you’d say, well, what else do I not know about his condition that I would want to know?”