1. A being with higher consciousness possesses two abilities A) the ability to discern between the object and a representation of the object, and B) the ability to apply concepts and form judgments about objects.
2. If a being has the ability to discern between the object and a representation of the object, and the ability to apply concepts and form judgments, then that being must be able to grasp the difference between the self and not-self.
3. A being is omnipresent when that being occupies or is present in all places, far or near, in all times, past, present, or future.
4. There is nothing that is not-self for an omnipresent being by definition of omnipresence.
5. So an omnipresent being cannot grasp a difference between the self and not-self.
6. Therefore, an omnipresent being cannot possess higher consciousness.
7. In short, God cannot have a mind because omniconsciousness is impossible.
(4) is obviously the weak point. The definition of omnipresence for the purposes of this argument is given in (3), but what about this definition gives us the claim that "there is nothing that is not-self for an omnipresent being"? The basic idea here is that an intelligently conscious being ("a being with higher consciousness") must be able to distinguish the objects of which it is conscious precisely as objects of consciousness. So in order to have a conscious being there must be a way to draw a distinction between subject and object -- self and not-self. This seems right enough. Where (4) seems weak is that getting (4) from (3) requires us to say that the subject/object distinction is an internal/external distinction: there is nothing external to an omnipresent being, so there is nothing that is not-self for the omnipresent being. As McCormick says:
If there is nothing external to a being or nothing that the being can accurately think of as external, then that being cannot draw a distinction between itself and objects which are not itself. There are no objects that would make such a distinction possible. Without the subject/object distinction, a being cannot possess either of the capacities of higher consciousness.
It's not so clear, however, that it is necessary to think of something as external, in a physical sense (i.e., the sense relevant to presence in a place), in order to recognize it as an object: we surely have internal objects as well as external objects. I not only recognize that I am not a fish, I also recognize that I am not my idea of a fish. I know I am distinct from the stone, which is external to me, but I also know I am distinct from the thought of the stone, which is not. Hume argued that the taste of the fig is not actually external to us: the taste of the fig really exists, but it doesn't exist anywhere. It has no location, although, of course, we can associate it with one through habits of imagination. If Hume were right about this, though, it wouldn't rule out the taste of the fig being an object of consciousness: it would merely tell us that something can be an object of consciousness that really exists but doesn't exist in any location external to us. And while one might not think the taste of a fig an especially good example, there are other examples in which Hume's claim is a bit more plausible -- e.g., desires. Likewise, in the course of making this argument, Hume also notes (and rejects) a position that, while very different, would have the same effect of prying the two distinctions (subject/object and internal/external) apart: namely, the scholastic doctrine that some things, e.g,. the soul, can exist wholly whole in many parts.
The point of bringing these up is not to insist that (say) Hume was right, but rather to point out that we seem to be able to make some sense of subject/object distinctions that do not rely on any distinction between the internal and the external, and which, if they really do occur, would still allow one to distinguish one's self from what is not one's self.
In the case of an omnipresent being are there any distinctions on the basis of which such a being could recognize the things it is present to as not itself? Certainly, and one of the more obvious ones really does follow from the definition of omnipresence in (3): the omnipresent being is omnipresent, whereas at least many of the things it is present to are not. An omnipresent being is present to, and not external to, a chair, but it is also present to, and not external to, a table, even though both the chair and the table have to be external to each other. So there is a distinction in mode of presence (God is present everywhere, chairs are not) that could perfectly well serve for a distinction between self and not-self.
Further, omnipresence, like most of the basic attributes attributed to God, is closely associated with the notion of God as cause -- in Christian terms, God as Creator and Provider. God's presence to everything is at least partly due to His causal relationship to everything. And thus there is a distinction here, as well, that could serve as the foundation for a distinction between self and not-self.
And, third, there are potentially differences in intrinsic character that could equally serve to ground such a distinction: for instance, God is simple, i.e., not composed of parts, while things that exist in bounded places typically are composed of parts. Here, too, we have a distinction that could distinguish the omnipresent being from the non-omnipresent being.
The basic Kantian idea at the core of this argument is that our ordinary self-consciousness of ourselves as existing in time (the temporal feature is actually fairly important for Kant's argument) requires being able to distinguish our selves from objects in space outside us. Kant holds that all our objects of consciousness are experienced under the forms of space and time, so in Kantian terms it would make sense to take, in our case, the internal/external distinction and the subject/object distinction to be closely related. But Kant himself recognizes that this need not be the case for every kind of consciousness -- indeed, he explicitly mentions God as an exception, since he thinks God is conscious of things not under the forms of space and time but simply in themselves. Further, since Kant's concern is refuting idealism, which denies that there is a real 'external' at all, his actual argument presupposes that there is at least a conceptual difference between the subject/object distinction and the internal/external distinction: because we can distinguish subject and object (and also recognize ourselves as persisting through time) it is necessary to say that there are objects external to us.
In other words, not to go on about this, the real question in response to (4) is, "Are there other distinctions beside the internal/external distinction that could serve as a ground for a distinction between self and non-self?" And it seems that there are many: philosophers have at least proposed some such distinctions in cases that don't even have anything to do with this subject, and an omnipresent being has certain features that already distinguish it from what is not omnipresent.