After all, he said, Medal of Honor recipients accept on behalf of their entire teams, especially those that did not return from battle. And canines have long been part of war efforts.

“This recognizes the dog is part of that team of brave people,” he said.

Mr. McCloughan, who taught high school and coached youth sports after his military service, added that he had worked with a dog in Vietnam who helped detect enemy activity during missions.

“They are very courageous,” he said of military canines.

He was also concerned about the condition of the dog, after the president tweeted that the dog was injured when the terror leader detonated a suicide vest.

“Once a medic, always a medic,” Mr. McCloughan said.

Mr. Trump on Wednesday also awarded the Medal of Honor to Matthew O. Williams, an Army master sergeant, who is credited with helping to save the lives of four critically wounded soldiers in 2008 in Afghanistan.

The image Mr. Trump shared on Wednesday was one of several altered photographs featuring the dog and posted in recent days on social media by The Daily Wire.

Mr. Trump took note of the Daily Wire photo mashup in a tweet he posted early Thursday, calling it “very cute” and suggesting that Conan was White House-bound next week.