In a telephone interview, Boyle sought to play down the uproar and said some crew members had made the patches a couple of years ago. "It's kind of old news," he said.
Spokesmen for the Army and Air Force, wanting no part of the brewing patch flap, referred all questions about their own policies on unofficial insignia to the Navy.
It is not the first time that Trump, while visiting US troops overseas, has drawn accusations that he is playing politics with the military.
During his surprise visits to US service members in Iraq and Germany in December, Trump singled out red "Make America Great Again" caps in a sea of military fatigues, signed a "Trump 2020" patch, and accused House Speaker Nancy Pelosi and other leading Democrats of being weak on border security.
Trump treated his appearance aboard the Wasp as a Memorial Day event because it was still Monday in the US when he spoke to hundreds of sailors and Marines, many shouting, "USA! USA!"
In the back of the buoyant crowd, some of the crew members wearing the Trump patches pleaded with news media photographers to get the president to sign their mementos. Trump was already signing dollar bills, Bibles and other souvenirs.
The proliferation of Trump hats and other paraphernalia at military events that Trump attends shows, in part, that rank-and-file troops feel more comfortable showing their political stripes, said Peter Feaver, a political-science professor at Duke University.
"Soldiers who would carefully avoid political speech in public would be cavalier about politicised posts on Facebook, Instagram or Twitter," Feaver said.
While difficult to document precisely, Feaver said it was likely that Trump's taboo-busting approach to civil-military norms was playing a role, one that some experts say could have a corrosive effect on that nation's armed forces.
Still, Feaver noted that the military has survived worse over the years: "The rank-and-file have always balanced a life of discipline, rigor and boredom, with transgressive acts as a way of letting off steam."
Written by: Eric Schmitt
Photographs by: Erin Schaff
© 2019 THE NEW YORK TIMES