"I need a formal order to act," Navy Secretary Richard Spencer said, and referred to the tweet. "I don't interpret them as a formal order."
Trump insisted last Thursday the Navy "will NOT be taking away Warfighter and Navy SEAL Eddie Gallagher's Trident Pin," inserting himself into an ongoing legal review of the sailor's ability to hold onto the pin that designates him a SEAL.
The Navy on Wednesday notified Chief Petty Officer Edward Gallagher that he will face a review early next month to determine if he should remain on the elite force.
Gallagher was acquitted of a murder charge in the stabbing death of an Islamic State militant captive, but a military jury convicted him of posing with the corpse while in Iraq in 2017. He was then demoted to chief.
Spencer, speaking on the sidelines of the Halifax International Security Forum in Canada, said if the president requests the process to stop, the process stops.
"Good order and discipline is also obeying the orders of the President of the United States," he said.
Despite the differing views with the president over the appropriate handling of the case, Spencer told reporters that he has not threatened to resign over the issue. But he acknowledged that he serves at the pleasure of the president.
"The president the United States is the commander in chief. He's involved in every aspect of government and he can make decisions and give orders as appropriate," he said.
Gallagher's lawyers have accused the Navy of trying to remove the SEAL designation in retaliation for Trump's decision last week to restore Gallagher's rank.
Gallagher filed a complaint with the inspector general accusing a rear admiral of insubordination for defying Trump's actions. Rear Adm. Collin Green is the Naval Special Warfare commander.
Under the review procedure, a five-person board will convene Dec. 2 behind closed doors. It will include one SEAL officer and four senior enlisted SEALs, according to the two U.S. officials. Gallagher can appear once before the board on Dec. 4 but without his lawyers. He can dispute the evidence given to the board that will include his conviction and call witnesses.
Gallagher can appeal any final decision that will be made by the Naval Personnel Board, which will take into account Green's input and the board's recommendations.
Trump's initial order in Gallagher only referred to restoring his rank, but it did not explicitly pardon the SEAL for any wrongdoing.
Green also notified three SEAL officers who oversaw Gallagher during the deployment — Lt. Cmdr. Robert Breisch, Lt. Jacob Portier and Lt. Thomas MacNeil — that they are also being reviewed, according to the officials.
Removing their Trident pins means they will no longer be SEALs but could remain in the Navy.
The Navy has revoked 154 Trident pins since 2011.