Why is the separation of Church and State a violation of the Constitution?

It's the opposite. The mixing of church and state is said to be a violation of the constitution. The first amendment guarantees that the US Government won't 'establish' a state religion. It's referred to as the establishment clause. And freedom of worship won't be abridged.

President Thomas Jefferson wrote the Danbury Baptists Association in the state of Connecticut in 1801 and stated:

"Believing with you that religion is a matter which lies solely between Man & his God, that he owes account to none other for his faith or his worship, that the legitimate powers of government reach actions only, & not opinions, I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should "make no law respecting an establishment of religion, or prohibiting the free exercise thereof," thus building a wall of separation between Church & State."