Unfortunately, this opinion is at odds with virtually the entire scope of 20'th century First Amendment jurisprudence. Not without reason, either - by this logic, your local police are perfectly able to suppress your First Amendment right to free speech, freedom of religion, and so forth, since the First Amendment is only binding upon Congress, and not them.
In any case, perhaps the appropriate next step is for supporters of Judge Moore to begin the process of amending the Constitution in order that it might more accurately reflect what they see as its rightful meaning in these sorts of cases...
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.