Greenwald isn't subject to the regular civilian justice system, rather he's dealing with the UCMJ (Uniform Code of Military Justice). That's what he agreed to when he voluntarily enlisted in the military. I'm guessing that they are going to charge him with treason which is a capital offense and thus he COULD get the death penalty, but that's hugely unlikely given the political skew and publicity of the case. I do, however, imagine he's going to spend a few decades in Leavenworth. And I'm okay with that.
Interestingly, Austrailia's law is about the same, except the max penalty is life in prison, not death.