An explosive report accused President Trump of making repeated disparaging remarks about members of the U.S. military, including those who died in battle, as “losers” and “suckers.” The report, published late last week by the Atlantic, relied on several anonymous sources to paint an image of a commander in chief who lacks respect for, if not openly disdains, American service members.

Trump has vehemently denied the allegations and accused the historic magazine of publishing a “fake story.” Several people close to the president have also gone on the record rebutting...