Displaying a replica of the Ten Commandments on public property doesn't violate the Constitution of the United States – regardless of public belief, common perception, or media reports. Rules about displays have come about over the last couple of generations.
The Supreme Court of the United States has issued a number of decisions concerning displaying a replica of the Ten Commandments in a public setting. Relevant pronouncements began with a case in 1947. As a layperson, it seems unlikely that the Supreme Court will revisit any major issues since decisions in 2005.
The issue begins with Americans trying to understand and live by the first amendment of the Constitution.
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof. . . .
These two phrases are popular references to the first phrase of the first amendment. The framers of the Constitution experienced the tyranny of living under an established religion. The Anglican Church was the official religion of England. England was was not especially different from lands in Europe. The Roman Catholic Church was the official religion throughout Eastern and Central Europe.
An established religion has a heavy say so in public life, even to the extent of vetoing laws passed by a parliament, for example. Other religions or denominations occupy precarious positions. Adherents may experience periods of toleration and then periods of persecution. The religious and political climate often follows the whims of the established religion.
Mike Schaps writes —
One may sketch the standard [of the Supreme Court's Establishment Clause] as follows: Government cannot favor one religion over another, or act to benefit religion over nonreligion, unless a government practice promotes nonsectarian religion only slightly and is so deeply woven into our national traditions that enjoining it would be highly divisive. Although this standard may appear unworkably vague at first glance, decades of Supreme Court Establishment Clause cases sufficiently calibrate it so as to make it judicially workable. That is, it provides adequate guidance to lower courts in all but a relatively narrow range of Establishment Clause cases. (p. 1244)