Spokeswoman Sarah Sanders said the White House believed there was a "legitimate reason and basis for not answering those questions".
"The President's throat was dry, nothing more than that", Sanders said, criticizing questions about the incident a day earlier as "ridiculous".
Many watching thought Trump was slurring his words during his announcement on Jerusalem.
Modern presidents have been examined by military doctors every year or two, and Trump has not yet disclosed any doctors' visits while in office.
"The full physical that most presidents go through, that will take place at Walter Reed, and those records will be released by the doctor following that taking place", Sanders said.
At age 70, Trump was the oldest president elected to his first term.
The White House said Trump was suffering nothing more than a dry throat. But she testified to Trump's endurance.
"I really think that he is wearing a denture and that denture is either ill fitting or it is slipping out of his mouth and that is causing his slurred speech", Dr. Nancy Rosen, a dentist in New York City, told Inside Edition.
It is the first time the White House has committed to releasing records about the president's health, something his predecessors did regularly.