That's a hard question... I was a navy guy myself, and I love all my military service members and vets. To answer your question most will say the marines for very good reason. But if your looking for elite, nothing gets more elite then the navy seals, not even marine Corp special forces. Each branch has at least one " elite" special forces group, but the seal team takes it for the most badass.
All branches from the marines to the air-force to the med. Staff are equal to me because the bravery required for all of those positions is extreme and I couldn't possibly compare anything I've ever done to any of those.
The only answer to this question is the Marines. The reason that I say this is because Marines are always faithful and will not quit. In Vietnam, not a single Marine flinched when the battle came to them or never hesitated when they had to go to the battle. Let a stupid navy seal do that. The Marines will always rule!