The argument from beauty, it seems to me, can come in four varieties, each asking a different "why" question, and each claiming that the best answer entails the existence of a being like God.
1. Why is there such a property as beauty?
This argument is the aesthetic parallel to the standard argument from morality. For it to work, a distinctively theistic answer to (1) must be offered. Parallel to a divine command metaethics, one could offer a divine appreciation meta-aesthetics. I think this gets the direction of explanation wrong—God appreciates beautiful things because they are beautiful. Moreover, if what God appreciates does not modally supervene on how non-divine things are, then divine simplicity will be violated. A better answer is that beautiful things are all things that reflect God in some particular respect, a respect that perhaps cannot be specified better than as that respect in which beautiful things reflect him (I think this is not a vicious circularity).
2. Why are there so many beautiful things?
The laws of physics, biology, etc. do not mention beauty. As far as these laws are concerned, beauty, if there is such a thing, is epiphenomenal. So, it does not seem that a scientific explanation of the existence of beautiful things can be given. But, perhaps, a philosophical account could be given of how, of metaphysical necessity, such-and-such physical states are always beautiful, and maybe then we can explain these entailing states physically. Or maybe one can show philosophically that, necessarily, most random configurations of matter include significant amounts of beauty, and then a statistical explanation can be given. But all that is pie in the sky, while a theistic explanation is right at hand.
3. How do we know that there is beauty?
This is parallel to my favorite argument from morality—the argument from moral epistemology. As far as naturalistic theories go, beauty (like morality) is causally inefficacious. As such, it appears difficult to see how we could have knowledge-conferring faculties that are responsive to beauty. The best story is probably going to be something like this. There is some complex of physical properties which correlates with being beautiful, and for some evolutionary reason, we have a faculty responsive to that complex of physical properties, and hence to beauty. However, the "and hence to beauty" is to be questioned. Evolutionary teleology is tied to fitness. The connection to beauty is fitness-irrelevant, because beauty is naturalistically causally inefficacious. It is at most that complex of physical properties that we are responsive to. But then it is not beauty as such that we are responsive to, that we perceive. Maybe, though, what we perceive is the most natural (in Lewis's sense) property among those that we could be reasonably be said to have states covarying with. But the physical correlates are, presumably, also quite natural since having states covarying with them is of evolutionary benefit. Moreover, I deny that the evolutionary teleology should snap to the most natural states, if the most natural ones are evolutionarily irrelevant. All in all, I do not think the prospects for a naturalistic account of our knowledge of beauty are good. But a theistic account is easy.
4. Why do we have aesthetic sensations?
This is an interesting question, but it strikes me as yielding an argument that is distinctly weaker than (3), unless it is just a different way of formulating an aspect of (3) (namely, the aspect of asking how our aesthetic beliefs get their intentionality). Still, the question is puzzling. We see such a very wide variety of things as beautiful: some people, most sunsets, many clouds, sme plants, most jellyfish, most tigers, most galaxies, some proofs, some musical compositions, some ideas, etc. It is odd that there would be an evolutionary benefit from being responsive to these things. The more likely naturalistic story is that this is some sort of a spandrel, maybe a spandrel of our recognition of good mate choices. Note that this story undercuts the attempt to evolutionarily ground our knowledge of beauty—it makes for us having aesthetic sensations but not aesthetic knowledge. That's a problem. But I am also not sure that the wide variety of things we sense as beautiful has enough in common for there to be a plausible story. However, that only yields a God of the gaps argument (not that there is anything intrinsically wrong with that).