I agree that his centrist-civil religion approach is consistent with America's Founding (that America's public institution's presuppose a Supreme Being, and therefore supplications to such ought to be constitutional). However, I think the scholarly case made by such figures as Steven Waldman and Jon Meacham is more accurate. Here is Dilulio's thesis:
The truth, however, is that present-day America is blessed to be in religious terms pretty much what Madison and most of the other framers intended it to be. It is a godly republic with governmental institutions that (as Justice Douglas phrased it) “presuppose” monotheistic belief in the “Supreme Being” known to Jews, Christians, and Muslims as the God of Abraham. It is a godly republic that affords a special civic status to nondenominational and interfaith (God-centered) religious expression. It is a godly republic that respects, promotes, and protects religious pluralism: Methodists, Muslims, Mormons, and all other faiths are welcome. It is a godly republic in which both the Constitution and federal laws prohibit government from discriminating against citizens who profess no faith at all (atheists have the same constitutional standing as Anglicans) or who are actively, but peacefully, hostile to all religion or to all church-state collaboration (Americans United for the Separation of Church and State is no more or less entitled to tax-exempt nonprofit status than the National Association of Evangelicals).
In his book, Dilulio takes slight issue with Jon Meacham's thesis which is well summarized in an article by Meacham here.
However, American history suggests that allusions to faith in the political arena are part of what Benjamin Franklin called "public religion," a religion whose God is perhaps best understood as the "Creator" and the "Nature's God" of the Declaration of Independence. This was not the God of Abraham or God the Father of the Holy Trinity, but a more generic figure who made the world, is active in it through the workings of providence, and will ultimately judge how people conducted themselves in life.
Taken together, the past reveals that the benefits of faith in God in our public life have outweighed their costs. "The sacred rights of mankind are not to be rummaged for among old parchments or musty records," said Alexander Hamilton. "They are written, as with a sunbeam, in the whole volume of human nature, by the hand of the divinity itself, and can never be erased or obscured by mortal power."
The issue is whether the God of the American Founding is the "God of Abraham." I would argue not necessarily, instead of simply not. The God of the American Founding is the God of natural religion [i.e., laws of Nature and of Nature's God], one whom all good men worship, regardless of whether their religion is Abrahamic. To Christians, it is a Triune God, to Jews, Unitarians and Muslims, it is a unitary father, and to others it is simply a Providence that goes by many names to many peoples. For instance, to Native Americans such Providence goes by the name "The Great Spirit."
As Presidents, Washington, Jefferson, and Madison repeatedly made public supplications to "The Great Spirit" by name, when speaking to unconverted American Indians. For instance, Washington:
I now send my best wishes to the Cherokees, and pray the Great spirit to preserve them.
-- TALK TO THE CHEROKEE NATION, August 29, 1796.
I now sincerely wish you a good Journey and hope you may find your [families and] Brothers well on your Return, and that [the Great Spirit above] 55 may long preserve your Nations in peace with each other and with the United States.
-- To THE CHIEFS AND WARRIORS, REPRESENTATIVES OF THE WYANDOTS, DELAWARES, SHAWANOES, OTTAWAS, CHIPPEWAS, POTAWATIMES, MIAMIS, EEL RIVER, WEEAS, KICKAPOOS, PIANKASHAWS, AND KASKASKIAS, November 29, 1796.
I receive with great satisfaction the visit you have been so kind as to make us at this place, and I thank the Great Spirit who has conducted you to us in health and safety. It is well that friends should sometimes meet, open their minds mutually, and renew the chain of affection. Made by the same Great Spirit, and living in the same land with our brothers, the red men, we consider ourselves as of the same family; we wish to live with them as one people, and to cherish their interests аз our own.
-- To the Brothers and friends of the Miamis, Pottawatomies, and Weeauks, January 7, 1802.
I thank the Great Spirit that he has conducted you hither in health and safety, and that we have an opportunity of renewing our amity, and of holding friendly conference together. It is a circumstance of great satisfaction to us that we are in peace and good understanding with all our red brethren, and that we discover in them the same disposition to continue so which we feel ourselves. It is our earnest desire to merit, and possess their affections, by rendering them strict justice, prohibiting injury from others. aiding their endeavors to learn the culture of the earth, and to raise useful animals, and befriending them as good neighbors, and in every other way in our power. By mutual endeavors to do good to each other, the happiness of both will be better promoted than by efforts of mutual destruction. We are all created by the same Great Spirit; children of the same family. Why should we not live then as brothers ought to do?
-- To The Brothers of the Delaware and Shawanee Nations, February 10, 1802.
I have a further advice of my Red children. You see how the country of the eighteen fires is filled with people. They increase like the corn they put into the ground. They all have good houses to shelter them from all weathers, good clothes suitable to all seasons; and as for food, of all sorts, you see they have enough and to spare. No man, woman, or child, of the eighteen fires, ever perished of hunger. Compare all this with the condition of the Red people. They are scattered here and there in handfulls. Their lodges are cold, leak, and smoky. They have hard fare, and often not enough of it.
Why this mighty difference? The reason, my Red children, is plain. The white people breed cattle and sheep. They spin and weave. Their heads and their hands make all the elements and productions of nature useful to them.
It is in your power to be like them. The ground that feeds one lodge by hunting, would feed a great band by the plough & the hoe. The Great Spirit has given you, like your white brethren, good heads to contrive, and strong arms, and active bodies. Use them like your white brethren of the eighteen fires, and like them, your little sparks will grow into great fires. You will be well fed, dwell in good houses, and enjoy the happiness for which you, like them, were created. These are the words of your father to his red children. The Great Spirit who is the father of us all, approves them. Let them pass through the ear in to the heart. Carry them home to your people; and as long as you remember this visit to your father of the eighteen fires, remember these are his last and best words to you!
-- To My Red Children, August 1812.
John Adams may well have done the same. However, I haven't been able to find his quotations. He certainly believed all world religions worshipped the same God and noted to Jefferson that Hindus worshipped the same God they did.
America's civil religion obviously presents a problem for atheists who don't believe in a God or for polytheists who don't like the supplication to a singular monotheistic God. However, America's civil religion may equally impose a philosophical problem for honest orthodox Trinitarian Christians who realize that all of these religions really don't worship the same God that they do and that America's Founders therefore erred in trying to construct a civil religion based on natural religion that held all good men of all religions worship the same God. See for instance Joe Carter's case against the civil religion here, where he also notes the idea comes from Rousseau, who was explicitly anti-Christian. America's key Founders including Jefferson, I don't believe consciously followed Rousseau. Rather, they seemed to absorb his powerful ideas through osmosis. But it doesn't change the fact that their civil religion (which is represented today by things such as "under God," "in God We Trust" and the National Day of Prayer) is a Rousseauian notion at its heart.
I don't think such an honest orthodox Christian should mind saying things like "under God" or "In God We Trust." After all, to him, these things can mean his own God. Just as long as he takes such with a grain of salt and understands the way America's Founders intended the civil religion to work was that generic references to "under God" likewise included concepts of God (like Allah or the Great Spirit) that he would consider false teachings.
Bottom line for orthodox Christians, don't look for redemption in politics; if you do, you will invariably commit idolatry against your God.