The Navy had been instructed by the White House military office during the trip to obscure the identity the vessel, named for the late US senator John McCain.
After details of the order became public, Patrick Shanahan, Mr Trump’s acting defence chief, is now considering sending out formal guidance to military units in order to avoid similar problems in the future.
Mr Shanahan confirmed details about a Navy email that said the White House wanted the USS John McCain kept “out of sight” of Mr Trump. The internal message came to light last week, triggering a storm of outrage.
Mr Trump, who refused to halt his long and bitter feud with Mr McCain even after the senator's death in August 2018, claims he had no idea about the directive.
However, the president later commented: “somebody did it because they thought I didn't like him, OK? And they were well-meaning, I will say.”
The order to hide the Navy destroyer reflected what appeared to be an extraordinary White House effort to avoid offending an unpredictable president known for holding a grudge.
Mr Shanahan told reporters travelling with him to South Korea on Sunday that he is not planning to seek an investigation by the Pentagon's internal watchdog into the matter “because there was nothing carried out” by the Navy.
He added that he still needs to gather more information about exactly what happened and what service members had done.
“How did the people receiving the information — how did they treat it,” Mr Shanahan said. “That would give me an understanding on the next steps to take.”
Mr Shanahan did not detail what those steps could be, but a defence official said Shanahan is considering a clearer directive to the military about avoiding political situations.
According to his spokesman, Lt Col Joseph Buccino, Mr Shanahan told his chief of staff on Friday to speak with the White House military office “and reaffirm his mandate that the department of defence will not be politicised.”
Asked directly if members of his senior staff were aware of the White House request before the president's visit, Shanahan said he’s been told they did not know.
He also has said he was not aware of the request and that he would never have authorised it.
What is still unclear, however, is who at the Pentagon may have known about the request and either agreed with it or chose not to discourage it.
It’s also not clear whether Navy leaders deliberately chose the McCain crew as one of the ships to be on holiday leave during Mr Trump’s visit, or if other measures were taken to ensure that the McCain was not visible from where the president stood when he arrived on the USS Wasp to make remarks.
The warship, commissioned in 1994, was originally named for the senator's father and grandfather, both Navy admirals named John Sidney McCain. Last year, the Navy rededicated the ship to honour the late senator as well.
Additional reporting by AP
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