I misquoted Mitt Romney in his belief that America had a divine founding. He used the word "inspired." It's a little milder, but official Mormon teaching uses the terms interchangeably. Here's a Mormon take on America's unique founding:
While LDS scripture reinforces the traditional Christian duty of "respect and deference" to civil laws and governments in general as "instituted of God for the benefit of man" (D&C 134:1, 6), Latter-day Saints attach special significance to the Constitution of the United States of America. They believe that the Lord "established the Constitution of this land, by the hands of wise men whom [he] raised up unto this very purpose" (D&C 101:80). The Prophet Joseph Smith once described himself as "the greatest advocate of the Constitution of the United States there is on the earth" (Hc 6:56-57). All of his successors as President of the Church have reaffirmed the doctrine of an inspired Constitution. This consistent endorsement is notable, for basic LDS teachings are far removed from the premises of American liberalism, and largely as a result of these differences, Latter-day Saints suffered considerable persecution before achieving an accommodation with mainstream America...
This understanding of the divine inspiration of the Constitution as mediated through the human wisdom of the founders and the founding generation invites the inference that new needs and circumstances might require the continued exercise of inspired human wisdom by statesmen and citizens alike. LDS leaders have taught that the Constitution is not to be considered perfect and complete in every detail (as evidenced most clearly by its accommodation with slavery, contrary to modern scripture; e.g., D&C 101:79) but as subject to development and adaptation... President Brigham Young explained that the Constitution "is a progressive — a gradual work"; the founders "laid the foundation, and it was for after generations to rear the superstructure upon it" (JD 7:13-15).
Two thoughts. The first is that this allows for far more constitution amending than most conservatives favor (and helps explain Romney's ease with amending the federal constitution to bar gay unions). Second: the question of American exceptionalism gets somewhat more dangerous when it is viewed as divine exceptionalism. In a world where the clash of Islamic and Christian civilization seems close to the surface, a reaffirmation of the divine nature of one nation (with the implication that all others do not enjoy the same divine favor) is not exactly oil on troubled waters.
For a Mormon Elder's view of the divinely inspired Constitution, check out this interesting article. The LDS church is particularly taken with the liberalism of the American constitution, especially its avowal of religious freedom.