Adam, my heart goes out to you. Please try not to feel shame or guilt over this. Your remorse and concern shows that you're true to your beliefs, even if you weren't able to reflect them in your actions this particular time.
I have not been in military combat but I'm from a military family and have many friends in the service. I have experienced some truly terrible things having lived in a combat zone, and heard even worse since then. What I can say is that you can't keep silent about this. It's good that you posted here, but I think you should also find some sort of support group specifically focused on military staff and combat trauma; I get the sense that you're feeling fairly isolated and alone in this right now. It can be hard for your heart to acknowledge this, but at least try to know, mentally, that you aren't the only person who's gone through this kind of hardship.
It's probably uncomfortable, and I know you said you didn't want to talk to people in Iraq about it, but what you need to do the most is talk face-to-face with another human being about this. Do you have someone in your unit that you can trust, or maybe a CO outside your unit that you have a close rapport with? I know talking to your own commander about this is probably the last thing you want to do, but if you're having nightmares and coming back to the same thoughts over and over, that's a liability to your unit and the people responsible for you need to know.
It's not your fault that you were put into this situation. While you could have not pulled the trigger, you did. It may have been a mistake, but I don't blame you for trying to do your job and protect yourself and your comrades. Your commanders are responsible for making sure that you are properly equipped, trained and briefed for your mission, and it's their job to take care of you. If you conceal your emotional condition now, you're not only preventing them from doing their duty, but you're also putting your unit and the people you're there to serve at risk. Most importantly you're putting yourself at risk. If you're not well, how can you expect to do right by the people that love and depend on you?
I know the military culture gets in the way of asking for help, and that likely a lot of the guys you work with have closed off their emotions or made sport of the situation. That's probably very alienating for someone of your background. But try to remember that that's just the way that they've learned to cope, from the cultures that they were raised in. It may seem very disillusioning and cold, but they're all just trying to survive a very ugly experience.
Remember that you, your brothers, your commanders, and even the civilians and the enemy all are just there trying to live on. You will see the worst of this conflict - corruption and hatred and desperation. The human animal isn't designed with warfare in mind, and it's hard to see past the horrors of combat. Try to remember that, at least at some level, you're there to help. Life is not something we want to compromise on, but occasionally you must kill in order to prevent your own death. You can do the most by living on.
The world is a harsh and lonely place; sometimes we're placed at odds with one another. I'm sure you're familiar with this saying; life is suffering. But you can curb your pain by learning from the experiences of others. You should really speak with someone in person about this if you can possibly muster the strength. Ask for a hug; try to remember what it means to feel the warmth and heartbeat of another person. That's why you're here, isn't it?
If you need someone to talk with in private about this, PM me. I'll give you the email-to-text bridge for my cell phone. I'm not always available, but I sleep poorly anyways so you can send a message any time without worrying about disturbing me.
Please don't doubt yourself. I know you're still on the path.
~Joe