In that thread I discuss, from a lawyer and legal historian's perspective, the senses in which these sorts of claims about the Judeo-Christian connection to U.S. law are false, as well as the senses in which they can be said to be are true. (I note that the impetus for that thread was a claim regarding the Ten Commandments specifically, but the thread wasn't limited to that.)
In a more general political sense, these sorts of arguments often hinge on how direct or indirect a connection one cares to draw. Just for example, a person might allow that the United States was founded upon Lockean principles. But Locke grounded his political philosophy largely in Christian theological principles as he understood them. Does this mean that the United States was indirectly founded, to some extent, upon Christian theological principles? Regardless of the answer to that question, it's kind of difficult to see what immediately practical policy implications would emerge from it.