If I want to recognize a kernel of truth the promoters of America's "spiritual heritage" tours have, it is, America, during its founding, embraced "religion" in the general sense and thought such provided "republics" with indispensable support. America's "organic law" is thus more religion friendly than a strict secularist ideal would allow for. However, whenever you start mixing religion and government, political/theological problems inevitably emerge. Key to founding thought was that all men of all religions had equal rights of conscience. Thus, whatever privileges government grants, it must grant equally without regard to religion. You want to fund private religious schools with tax dollars, then Islamic schools are equally entitled to such aid, as long as they meet the secular criteria.
George Washington struggled with the political theological problem when he fought for John Murray -- a universalist who denied the existence of eternal damnation -- to be a chaplain, over the protests of the more orthodox chaplains. [Benjamin Rush likewise converted to this creed of Trinitarian Universalism.]
Washington's own view of the afterlife is hard to pin down. He certainly believed in it, but often expressed opinions about such in non-traditional Christian terms. For instance, in one letter he cites a pagan source -- Cicero -- as authority for existence of the afterlife.
But with Cicero in speaking respecting his belief of the immortality of the Soul, I will say, if I am in a grateful delusion, it is an innocent one, and I am willing to remain under its influence.
This is not exactly how an orthodox Christian would put it. Personally I believe that Washington, like the other key founders probably believed that good people by their works merit Heaven immediately upon death, the bad, temporarily punished, eventually redeemed. Washington's writings give no hint that he believed in eternal damnation. Or if he did, he certainly had a cavalier attitude towards the concept. The orthodox who posited the notion of eternal damnation tended to think it very important that folks believe it and termed theological universalism "infidelity." For instance, Bishop Meade, an Episcopalian and an orthodox Christian, stated: “I have other reasons for knowing that infidelity, under the specious garb of Universalism, was then finding its way into the pulpit.”
Evangelicals today seem to have the same attitude. Evangelical minister Carlton Pearson denied eternal damnation and lost his entire ministry after being labeled a "heretic." George Washington on the other hand, praised a church that preached this very "heresy." Washington made it clear that whatever it was he valued about religion, the “infidel” Universalists had it. That's one reason why I think Washington himself disbelieved eternal damnation.