America functions within a moral framework, meaningful only when attached to a Creator. To quote George Washington in his farewell address, where he warned his country against any attempt to build a moral framework apart from God:
"Of all disputations and habits that lead to political prosperity, religion and morality are indispensible supports. In vain would that man claim the tribute of patriotism who should labor to subvert these great pillars of human happiness-These finest props of duties of men and citizens...And let us with caution indulge the supposition that morality can be maintained without religion. Whatever may be conceded to the influence of refined education on the minds of peculiar structure, reason and experience, both forbid us to expect that national morality can prevail in the exclusion of religious principle."
And the framers of the Declaration of Independence certainly did NOT, when they referred to God, envision the theocracy of Islam, where religious belief is a compulsion.
You're right that morals matter. And in many cases, those morals must be imposed.