God is not the property of any church, Sorcha. ;-) (though they mostly seem like to imagine that they have an exclusive franchise on God...)

You can separate the state from churches, certainly. But how can you separate the state from a concept which is not under the control or grasp or jurisdiction of churches, no matter how much they try to control or grasp it? God (at least in one sense) is an idea. It is an idea of something unlimited and transcendent that underlies our existence and which determines the nature of existence itself. In many cultures it is the idea of something omnipresent and eternal. Such ideas exist and they always have in philosophy, but they cannot be separated from a state or a society as long as any of the people in that state or society have those ideas resonating in their consciousness.

To invoke God in an oath is NOT to bring any particular church into the government, it is simply to invoke the idea OF God itself, and that idea exists outside of churches as well as within them. It has always existed both in AND outside of churches.

I belong to NO church, yet I do have an idea about God...my own particular idea about it...which came from no church that I know of...and if I were part of a government and I expressed some of that idea in a ceremony I would not be bringing "the church" into "the state", because I do not subscribe to "the church" in the first place. The church plays no part in defining my concept of God.

Furthermore, if there is anything like a universal principle that is 100% real and woven into the nature of our existence, and that could be termed "God", then by definition it pre-existed ALL churches, and as I pointed out above...it is NOT their property now nor is it under their jurisdiction...nor can it possibly NOT be part of the state, because everything that exists arises out of it in the first place.

What people are objecting to here is a form of outward ceremonial that they don't happen to subscribe to themselves, and they're saying "I don't believe in that form of ceremonial (or the philosopy behind it), and therefore I don't want other people to do it either."

Well, tough! Other people in this world are always going to do a few things in a ceremonial or formal sense which you don't choose to do, and if you don't like them doing it, don't WATCH it. Go focus on something else you like instead and stop trying to force other people to be the same as you. That's not their choice.

And who made you or anyone else the arbiter of what someone else's choice should be? Barack Obama would, I believe, choose to invoke God in his oath of office, and that's just fine for him.