Existence exists, and only existence exists. Existence is a primary: it is uncreated, indestructible, eternal. So if you are to postulate something beyond existence—some supernatural realm—you must do it by openly denying reason, dispensing with definitions, proofs, arguments, and saying flatly, “To Hell with argument, I have faith.” That, of course, is a willful rejection of reason.Vallicella points out the bad reasoning and highly questionable assumptions inherent in this "argument." I encourage others to read his post.
I would like to add to his points, though. I fear there's some confusion going on in Objectivist thinking on this matter. For example, Objectivist Leonard Peikoff writes,
We start with the irreducible fact and concept of existence – that which is. The first thing to say about that which is is simply: it is. As Parmenides in ancient Greece formulated the principle: what is, is. Or, in Ayn Rand’s words: existence exists. (“Existence” here is a collective noun, denoting the sum of existents.) This axiom does not tell us anything about the nature of existents; it merely underscores the fact that they exist. (Peikoff, Objectivism: The Philosophy of Ayn Rand, p. 4, emphasis mine)So the fact that "existence exists" doesn't tell us anything about the nature of the existents means that Peikoff can't make a direct argument from this axiom to the claim that "existence exists" means that the "supernatural doesn't." If "existence exists" just means "only non-supernatural things exist" then it appears that "existence exists" does tell us something about the nature of the existents. Moreover, to say that the nature of the existents in "existence exists" is that they are "uncreated, indestructible, eternal" seems to be close to contradicting yourself when you say that "existence exists" tells us nothing about the nature of the existents.
This makes their atheological cash value close to nil.
We can add another term to the above. Since The "primacy of existence" is contrasted with the "primacy of consciousness" then it looks like "unconscious" should be added to the list, thus we'd have: "uncreated, indestructible, eternal and unconscious." So, once we break down the term "existence exists," it appears to fail to function as an axiom. Indeed, the claim is more loaded than a baked potato from Outback steakhouse!
I suggest that this is one reason why you will never see the argument from the primacy of existence in any reputable compendium of atheological arguments. As Vallicella said,
"If someone wants to argue that only physical objects exist, he is free to do so. But one has to produce an argument, not rig one's terminology in such a way that the existence of nonphysical objects is ruled out from the outset."
Yes, by all means argue that only physical objects exist; for instance, like the laws of logic you must employ in your argument. Or the mind that forms conclusions caused by other beliefs. Oh, and then try and make your case for normative ethical egoism on physicalist assumptions (that might be unethical considereing it might not be in a egoist's best interest to argue for ethical egoism over altruism, even if egoism is true, since the more altruists there are, the better it probably will be for you).