An interesting argument, but somewhat simplistic. The USA was founded by people following laws and guidlines springing out of Judeo-Christianity. Admittedly they weren't doing it very well, but the American constitution recognises this as a problem and seeks to separate church from State.

However, it does claim to be one nation under God. The judeo-christian influence on people like the American founding fathers, Isaac Newton or even Charles Darwin can't be undone - Christianity was a way of life and so it underpins the psychology of the people who did the creating of the laws.

And it's a mess. That's why you end up with people who are anti-abortion but pro-death penalty or vice versa. I'm not sure you can retrospectively take the Christianity out of either US or European society any more than you can take Buddhism, Hinduism or Islam out of Chinese and Indian and Middle Eastern societies respectively.

Much as I (and it seem you) would like to do.