The way I understand the book's thesis, it has to do with political-theology -- the old saying politics is theology applied. You don't need any kind of traditional or orthodox understanding of the faith. Rather, Rousseau's civil religion or ceremonial deism will do. But the "system" has to have some kind of divine trump.
The religious conservatives are right: there is a theology behind the American political system—only it isn’t Christianity. It’s deism, the faith most closely associated with the Enlightenment, which professes, as Critchley puts it, that “there’s a God, but a God that doesn’t do party tricks.” Even if no one calls himself a deist anymore, it lives on it the political systems that the Enlightenment inspired—especially our own. Liberal democracy, Critchley argues, is simply the political form of deism. Natural law and natural rights, so central to the American creed, are fundamentally theological concepts. Thomas Jefferson may have been a freethinking, Bible-revising iconoclast, but he wasn’t just being figurative when he wrote, in the Declaration of Independence, that such rights are endowed by a Creator; that’s what deists believe. And even without prayer in schools, the deist creed is coded into every national ritual we have, from the courtroom to the ballpark.
Is there any way to participate in politics without getting religious? “I don’t think it should even be an aspiration,” Critchley told me. “If you look at a counterexample, the problem with the European Union is that it doesn’t have those rituals. It tried to bind a polity together through a constitution, but it was so weak. There’s been a total failure to craft something like a European identity—the problem hasn’t even been recognized. So we’re left with a unity which is simply monetary. And that seems to be screwed.”
I'm not sure if I would call it "deism" as opposed to "generic monotheism." As I have come to learn the civil religion or political theology of the American Founding is an uber-eccumenical generic monotheism where among others, Jews, Christians, Muslims and un-converted Great Spirit believing Native Americans all worship the same monotheistic God. AND further, much of what has been termed "deism" from this era is actually Providential, heterodox (non-Trinitarian) "Christianity." (I put "Christianity" in quotes because, to some, non-Trinitarianism and "Christianity" are mutually exclusive concepts.)