There is a lively discussion at Church of the Masses on the question of whether God is Catholic. Most of the people who say He is argue on the basis of the truth of the Catholic Faith. Most of the people who say He isn't are upset at the perceived triumphalism of those who say He is.
Lane Core reduces his answer to a kind of syllogism:
God the Son, the Second Person of the Holy Trinity, is the Head of the Catholic Church. Therefore, God is Catholic.
This argument uses something like the "communication of idioms" principle by which we say things like, "God died on a cross."
Personally, I think the question, "Is God Catholic?" is extremely ill-formed, so much so that the unhesitating Yes!es I've seen make me suspect they're answers to a different question. Something like, "Is what the Catholic Church teaches about herself and her relationship to God true?"
And I would unhesitatingly answer that question, "Yes." But as far as I know -- and setting aside the communication of idioms for a moment -- in teaching about herself and her relationship to God, the Church doesn't teach that God is Catholic.
I'm not even sure what such a doctrine would mean. Does God have faith that what the Church teaches is true? Does God have hope in His salvation? Has God even been baptized in the name of the Father and of the Son and of the Holy Spirit?
Some of the difficulties with this question can be resolved by answering the related question, "Is my guardian angel Catholic?" You can shake out what it means for a spiritual being to "be Catholic" without getting into issues of what it means to say "God is" something.
But the certainty with which many Catholics assert that God is Catholic makes me wonder about how clearly Catholics think about God. If God is Catholic because Jesus founded the Catholic Church, isn't God also Jewish? In fact, wouldn't God have become Jewish at one point in time, and at a later time become Catholic (or Christian, for Protestants who don't care for Catholic triumphalism)? And wouldn't that mean, contrary to Catholic dogma, that God changes?
Any time you say "God is [something]," you're saying something pretty outrageous, even -- especially! -- if it's true, and you need to be very careful about it.
My biggest objection to saying, "God is Catholic," though, is that it betrays a perception of a limited God. The Church, that statement tells me, is a container in which can be found you, me, God, and a bunch of other people. I don't think Catholics really appreciate how utterly not-like-us God really is. There are no categories that contain both us and God. God is, God is Being, and not just the Supreme Being imagined as the top of a pyramid of beings or the far endpoint of a line of beings.
God's Being is not somehow more than ours; it is other than ours. When God told St. Catherine of Siena, "I am He Who Is, and you are she who is not," He wasn't saying he was 100% on the Being Scale, while she was 0%. He was saying she couldn't even be placed on the scale.
We speak of an infinite God, as though "infinite" were a positive attribute: just keep adding one and you'll reach infinity; just keep moving in this direction and you'll reach God. What it really means, though, is God has no limits. The mathematical concept of infinity is as a limit, almost the opposite of Divine infinity. And God not only isn't at the end of the number line, He's not above or beyond it either. Creation, and all the number lines it contains, doesn't exist in the same space as God, because God doesn't exist in a space.
Okay, divide the number of words used with the number of points made to get an idea of how well I can express myself on this. But there's one further thing to notice about all this: To the extent we fail to appreciate how unlike us God is, we fail to appreciate the magnitude of His free offer to join in His Triune life.