If, however, God exists, he is a living being. If he is a living being, he also possesses sensation. For it is precisely he fact of possessing sensation that differentiates a living being from what is not a living being. But if he possesses sensation, he hears and sees and smells and touches. And if this is so, there are certain things in the realm of each sense which attract or repel him....But if this is so, there must be certain things which are vexatious to God; and if there are certain things which are vexatious to God, God is subject to change for the worse, hence also to destruction. Therefore God is perishable. But this is in violation of what was the common conception of him. Therefore the Divinity does not exist.
Probably the most interesting feature of the argument is the assumption that sensation implies the possibility of vexation. That is to say, that for every sense there is necessarily something attractive and something repulsive (sweet and bitter, for instance); given that, and the claim that the gods must possess sensation (the basic reasoning there is that if there is a sense a god does not have, it will follow that human beings and other living animals are superior to that god in that respect), it follows that the gods can change for the worse. But this is inconsistent with what, for instance, the Stoics insist when they talk about God.
An equally interesting argument, which goes for the same artery:
If, however, the Divinity exists, it is in any case a living being. If it is a living being, it is at all events all-virtuous and happy (and happiness cannot subsist apart from virtue). But if it is all-virtuous, it also possesses all the virtues. But it cannot possess all the virtues without possessing both self-control and endurance. And it cannot possess these virtues unless there exist certain things which for God are hard to abstain from and hard to endure....And if there exist certain things which for God are hard to abstain from and hard to endure, there also exist certain things which can change him for the worse and cause him vexation.
At a more abstract level, the assumption underlying both of these arguments (and several others that Sextus Empiricus gives) is that life intrinsically carries the power to vex the living. If you are alive, then, necessarily, you can be worse off. The Stoics, who seem to be in the background here, would surely have denied this claim; for that matter, it seems to me that most schools would have -- it's as inconsistent with Platonic, Peripatetic, and even Epicurean conceptions as it is with Stoic conceptions. But at the same time Sextus Empiricus takes the assumption to be quite plausible -- he certainly uses it too often in too many different arguments to be unaware that he is using it. It does bring out quite well how important it would have been for them to develop an account of the unvexable life; the Stoics had started it, and certainly it would become a major Neoplatonic project.
Quotations from Sextus Empiricus, Selections from the Major Writings on Scepticism, Man, & God, Hallie, ed., Etheridge, tr. Hackett (Indianapolis: 1985) pp. 206-209.