No, it was an Enlightenment philosophy that was pretty much limited to Northwestern Europe and North America. That philosophy grew more from the horrors of 250 years of inter-Christian bloodletting than it did from scriptural teachings.
The men who developed it were not "anti-Religion" (most were deeply religious) but they understood from history and often from their own experience, that mixing religion and state was a recipe for oppression.
In the 18th Century, the "Christians" of Berlin, Moscow, Madrid, Budapest or Rome could not have produced the Constitution any more than the Saudis could do it today.
The Constitution was based upon philosophy, not dogma.
I've never said that Islam and Christianity have a similar worldview. In fact, if you look, I pointed out that for Muslims, all of reality is dependent upon the continual will of Allah, whereas for Christians, God created a rational, self-functioning Universe.
Now, socially, we are in total agreement, I think: this is a "Christian nation" for all intents and purposes. SOCIALLY. But LEGALLY, it is a country that protects the freedom of ALL religions, and all non-religions. Period. Christianity was not codified in the Constitution. Period.
Theology is not the issue here--citizenship is. You CAN be American and whatever religion you choose or don't choose. Maybe AdamSmithWasRight is not a "good" Muslim because he keeps his loyalty to his country, but that doesn't make him any less a citizen.