I don't think one can make an empirical case that religion is necessary or even particularly helpful for maintaining a morally upright population. (Unless one circularly argues that the morals that count are exactly those espoused by one religion.)

I do think religion played an important role in the morals of the founders themselves--they make that very explicit in their writings, but that's a different issue.

However, it is an interesting question whether religion can provide critical social cohesion in a land of immigrants without an extremely long cultural history.

Right now the answer seems to be "no, the more religions are among those most fervently engaged in the culture war which is fragmenting social cohesion".

But that is a slightly different question. The United States is not an exceedingly religious country any longer (though it's very religious for a country with a high level of technological and economic advancement), but one can ask: what if it were? Would the social cohesion be higher, and would that be enough to stop the fragmentation and polarization?

(Of course, if you specify not that there is simply religion but one particular religion, at some point there wouldn't be anything to fight about culturally because everyone would have the same narrow set of beliefs. But I don't mean that. I mean the kind of diversity you'd expect from a land of immigrants with separation of church and state...just one that was even more religious.)