Well, at a time before State and the American Church began that spiritually perilous trek, exposing divisions, chasms and splinters over many generations, you would have to be fully out of touch to think there has been no such impact on the founding to the present, formally or otherwise. Take a peek as 'Desk Trauma' reminds us:
The great, vital, and conservative element in our system is the belief of our people in the pure doctrines and the divine truths of the Gospel of Jesus Christ. US Congress 1854
[T]he only means of establishing and perpetuating our republican forms of government is the universal education of our youth in the principles of Christianity by means of the Bible. Benjamin Rush 1798
[T]he religion which has introduced civil liberty is the religion of Christ and His apostles… This is genuine Christianity and to this we owe our free constitutions of government. Noah Webster 1843
We all agree in the obligation of the moral principles of Jesus and nowhere will they be found delivered in greater purity than in His discourses. Thomas Jefferson 1809
The Christian religion is, above all the religions that ever prevailed or existed in ancient or modern times, the religion of wisdom, virtue, equity and humanity. John Adams 1796