I know it probably doesn't seem like it, but I'm not a Gallagher fan. In fact I think there's a good chance he killed that kid, but he was found not guilty of that at Courts Martial. The system failed whether he's guilty or not. There were unethical and illegal tactics used in an overzealous attempt to nail this guy. It would have been extremely difficult to get a conviction if they had not. Regardless you can't play that way. Civilian detectives and prosecutors can't use illegal means and violate peoples rights, even if they are sure an individual committed a crime. We wouldn't allow that there and we shouldn't allow it here.
Admiral Green was brought in to clean up NSW. Since the start of OEF/OIF NSW has gone further and further off the reservation to eventually become the biker gang of DOD. As a JAG friend at RLSO SW told me, NSW is notoriously impossible to prosecute. They put up so many road blocks and cling to "the brotherhood" to such a degree that normal prosecution is a very difficult task. Because of that, NSW imposed pressure and restrictions on Gallagher and his legal team from day one that would have been unjust for any other defendant. I have no doubt Admiral Green thinks he's guilty of murder. Now he's grasping at any straw he can to make sure that Gallagher is in some way punished. If posing with a dead fighter is so egregious (which it is) as to warrant taking his trident, then it warrants taking the trident of an entire platoon of ST7 that posed for that picture. But he can't do that without completely turning the teams against him. There is no good way forward.
On the other hand, the POTUS has told him to stop. A POTUS elected by the people (no I'm not a cultist). We have an elected civilian leader of the military for a reason. We can't just choose which orders to follow and which to disobey, even if we think that its out of that persons swim lane. We are in new territory on a daily basis with President Trump. But going down the road of military leadership defying the POTUS is a precedent that we don't want to set. To quote President Obama "That's not who we are."