Well, they would have their own constitutions to answer to.
My views on the 14th amendment are known, but I'll accept for the sake of argument it was intended somehow or another to refer to the BOR even though it makes no mention of it. We're still left with the actual words of the amendment - "privileges and immunities", "life, liberty, and property", "equal protection". Now the 1st amendment, on its face, is applicable only to Congress. That much is indisputable. The only way it can go on to apply to the states is through the filter provided by the aforementioned words from the 14th.
IOW, in order to make the case that Moore's actions are unconstitutional, one would have to show that he abridged anyone's privileges or immunities, or denied anyone life, liberty, or property without due process of law, or deprived anyone of the equal protection of the laws. Good luck.
Easier than you think. Judge Moore used the platform of the courthouse to promote his own personal beliefs, thus enjoying the liberty to do so, and yet denied an atheist organization the same liberty when they sought to place a display - a sculpture or picture of an atom, according to the trial court record - representing their own beliefs in the same courthouse rotunda. Thus, Judge Moore has promoted his beliefs, yet denied the freedom of others to do the exact same thing.
No more divine right of kings, please - Judge Moore does not have special liberties unavailable to everyone else. Such is the clear and explicit meaning of "equal protection under the law".