I don't remember where in any official document of the United States that it says which God. I do know that it was so important to the founders of this country that freedom of religion be expressed that they made it the first topic in the bill of rights.
What do the Ten commandments have to do with Christianity? Nothing save through an extension of Judaism. The Ten were given to Moses, a Jew, at the top of Mt. Sinai. So the basis of morality of the Christian faith are also the basis of morality for Islam, and their mutual progenitor faith, Judaism.
It is possible for Atheists to have good morals... but they didn't come to these by accident. The rules they follow were still given to them by God, whether they believe in him or not. Society has enforced the rules for morality, and up until this point, society has been rooted in faith. So even if there isn't a God (which there is), it is our faith in him that has given us our moral code.
Besides this point, all major religions of the world have some sort of moral code, and by the same processes, it is faith in their beliefs that have shaped morality in their societies. SO WITHOUT GOD (under whichever name you call him), THERE ARE NO MORALS.
Back to the first amendment....
We are granted the freedom "of" religion. Not the freedom "from" religion, however that is a choice. Separation of church and state doesn't permit the oppression of those wanting to practice their faith, but actually protects those wanting to do so. It is a shame that so many courts have interpreted this protection as a divisive method of oppression.