I sort of doubt that a U.S. soldier would get off scot-free for shooting up a Bible, either in Iraq or at a U.S. base. The U.S. is occupying a country where the vast majority of people - most of them Muslims of one stripe or another - want us to leave. The idea is that we're supposed to be helping them establish a stable democracy and be sensitive to their religious and cultural realities. Frankly, I think that's a lost cause when you invade a country and the primary mode of "caring" is bullets and bombs. People understandably fail to see a more nuanced message there. We're occupying THEIR country with our military. They want us to go. Separation of church and state is not their thing so much. I think it should be, but I don't think we can force that value on them militarily, and we're not succeeding in getting them to adopt that value (we're not respecting our OWN separation of church and state enough these days, so that's no surprise). So their religion and their politics basically merge together. Having a guy on our side shooting up a copy of the text recognized as holy by most of the citizenry seems rather unhelpful and unconstructive, regardless if one is for or against the war.
Also, let's not forget that it's NOT the United States Christian Army. It's the United States Army. There are plenty of Muslim Americans and plenty of Muslims serving in the U.S. Army (and the other branches).
I have mixed feelings about it, but as far as I know, signing up with the military does seem to involve a voluntary curtailment of some of your rights to free speech. Soldiers aren't as free to criticize the Prez as the rest of us, no? And I'm guessing they can't hope to get away with burning a flag, either. That one seems reasonable, all things considered. Shooting religious texts also strikes me as behavior that soldiers ought not be engaging in.
If I were in charge of a platoon (I know, hardy-har-har), I wouldn't allow any of my troops to waste bullets on such frivolous exercises of hate - or an incredibly juvenile and pointless stunt. Just shoot at a normal target, the enemy that's shooting at us, or just skip firing a gun. Taxpayers pay for those ridiculously overpriced bullets, remember.