I would argue that the actual Ten Commandments have had very little observable effect on American history. Religion in general has been extremely important in US history. However, the Ten Commandments themselves have not.
Of course, most Americans have always professed to be Christian. In addition, Christianity has had a great impact on our history. For example, it was the Christian “Second Great Awakening” that led to such reform movements as the abolitionist movement and the women’s rights movement of the antebellum era. It was Christian religion that had a large part in bringing about Prohibition. Christianity was a major basis for the Civil Rights Movement, which was led mainly by clergymen from the black community. In all of these ways and many more, Christianity and Christian values have had a great deal of impact on American history.
However, this is not the same thing as saying that the Ten Commandments had any specific major role in American history. Our history is full of people killing one another. It is full of people stealing. It is full of people lying. It is full of people committing adultery. It is full of people taking the name of God in vain. There have been places and times when it was hard to find an open store on a Sunday because of the commandment about keeping the Sabbath, but this is not a particularly important thing in the overall scope of American history.
For these reasons, I would say that Christianity has had a great impact on American history but that the Ten Commandments have not.