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".
Now, would you consider it a similar violation if he put up a copy of MLK's "I Have a Dream" speech, while at the same time refusing a request to put up an address by George Washington? Does he have to accept every request for a display that someone wants to put up, no matter how irrelevant it is to the historical foundation of the country's laws?