No really he doesn't. At least not by my definition of exist. Think of it this way. To exist is to stand (sistere) out from something (ex).* Existence is only possible within a relationship to something greater. All things that exist do so within a field of being that is larger than themselves. So the corn exists within the field, the field exists within the country, the country exists within the planet, which exists within the galaxy, etc, until we arrive at the expanding field of time and space that is our universe beyond the edge of which is what? A multiverse, other dimensions? And what do they exist in? Nothing that can be said to exist can be thought of as existing without relating it to something greater than it.
|Corn, in a field|
God doesn't exist, in that there is nothing in which God has God's being other God's-self. The universe and all that is in it exists only in that it is distinguishable from God and yet stands in relationship to God. God, whose name is revealed as YHWH (I am what I am, I will be what I will be) does not exist but simply IS. There is no other thing or person of which this can be said.
Let me know what you think :-)
*C.T. Oden, Classic Christianity, p35