I was wondering about that. I know even Americans who can't keep the two seperate.
On the other hand, note that at the beginning of the Declaration we have the phrase:
the separate and equal station to which the Laws of Nature and of Nature's God entitle them
Notice how they invoked Nature and Nature's God. That is a key that the god they were referring to was the Deist god, and similar to the anti-clerical nature worship that the leaders of the French Revolution tried to institute.
Toward the end we have:
And for the support of this Declaration, with a firm reliance on the protection of Divine Providence, we mutually pledge to each other our Lives, our Fortunes and our sacred Honor.
The phrase "Providence" is another code word usded by the Deists.
In short, the so-called religious phrases were just the usual boiler-plate rhetoric (the Declaration of Independence was, after all, polemic). I wouldn't say that religion or religious sentiment was a central theme of the Declaration of Independence either.
"The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one." -- George Bernard Shaw