The multiverse is certainly a very clever idea. However there are a few problems with using it to avoid these theistic arguments. Before I get into them, though, I'd like to make two points that aren't objections so much as interesting postulates. First, as I point out here, the multiverse can be used to obviate the argument that the occurrence of evil is incompatible with God's existence just as much as it can be used to obviate cosmological and teleological arguments. So if we use it to take away some reasons for believing in God, we can also use it to take away some reasons for not believing in God. Second, the multiverse hypothesis, if successful, would negate cosmological arguments based on the universe having a beginning and all teleological arguments. Yet these arguments have been around for millennia and I'm unaware of anyone employing a multiverse concept to get around them. Of course this doesn't mean it's false, but perhaps it should make us a little suspicious.
Anyway, here are the problems, as I see them, with using the multiversial to avoid theistic arguments.
1. The multiverse is just as metaphysical an explanation as the claim that "In the beginning God created the heavens and the earth." Appealing to the multiverse's natural processes in order to account for our universe's origin does not make it a physical explanation, since those processes transcend the processes of the matter, energy, space, and time that make up our universe.
2. No one has yet been able to produce a model for a multiverse that does not itself have a beginning. So it doesn't really remove the necessity of an ultimate cosmic origin, it just pushes it one step back.
3. At any rate, cosmological arguments did not originate with the discovery of the Big Bang. They have been defended for millennia based on the mathematical problems that arise if we posit an actual infinite amount of things. In order for the multiverse to not have a beginning itself, it would entail an actual infinite number of cause-and-effect events, and so the mathematical problems are still applicable.
4. Ockham's Razor plays havoc with the multiverse. This is the claim that we should prefer simpler explanations that posit fewer entities over complex explanations that posit more entities. Ockham's Razor is one of the most important principles in science. In order to account for one universe having the right conditions for life the multiverse posits trillions or an infinite number of other universes. In contrast, the theistic explanation requires us to posit one further level of reality to this universe. If we have to choose between these two options, the claim that God created the universe wins hands down.
4.1. It may be objected that the God being posited, as the creator of the universe, would be enormously complex, and so Ockham's Razor, which prefers simpler explanations, would point us to the multiverse. This, however, misunderstands two things: first, in Ockham's Razor, "simple" does not mean ontologically simple, it means numerically simple. To put it another way, it is not a matter of qualitative complexity but of quantitative complexity. The Razor claims that, all things being equal, we should prefer explanations which posit the fewest number of entities. The multiverse posits innumerable other universes in order to explain this one. Theism posits one other realm of reality in order to explain it. We should prefer the latter over the former according to Ockham's Razor. Second, traditionally the God of theism has been conceived as being the simplest of all beings. This is known, not very imaginatively, as the doctrine of divine simplicity. So, even if we ignore the first point, theism is not positing a more ontologically complex explanation of the universe than is the multiverse.
4.2. It may be objected further that the multiverse is not really positing all these other universes as distinct entities, but as outgrowths of a single all-encompassing ur-cosmos. There are two problems with this: first, we can do the same thing with the theistic explanation. Our universe is a part of reality; the whole of reality includes God and everything else he has created. As C. S. Lewis put it in Miracles, atheists "have mistaken a partial system within reality, namely Nature, for the whole." Second, at any rate, this is not a viable strategy, since any charge that something conflicts with Ockham's Razor could be explained away by saying all the other entities being posited are just parts of a larger singular entity. In other words, if we say that the multiverse doesn't conflict with Ockham's Razor, nothing else does either. Ockham's Razor is defunct and empty. This is not a reasonable conclusion.
5. In addition to flying in the face of Ockham's Razor, the multiverse commits the inverse gambler's fallacy. This plays on the much more famous gambler's fallacy. If someone sees a coin being flipped a hundred times and it comes up heads each time, he commits the gambler's fallacy if he bets the coin will come up tails on the next flip because he thinks it's due. The inverse gambler's fallacy says that, regardless of the merits of the bet, the gambler is essentially assuming that if there were innumerable coins being flipped, one of them was bound to come up heads a hundred times in a row. Yet this would only be a viable explanation if the gambler had actually witnessed all these other coins coming up with all their other results. Without such observation, you're best off thinking that the coin-flips are fixed somehow. Similarly, if we find ourselves in a universe that meets just the right conditions for life, we're best off thinking that the game is rigged: the universe was made that way on purpose.
6. The multiverse hypothesis, by itself, is not sufficient to avoid the cosmological and teleological arguments. We must specify a multiverse of a particular type and character. This is problematic because the more conditions one has to add to the bare-bones multiverse, the more contrived or ad hoc it is; and the more ad hoc an explanation, the less likely it is true.
6.1. Having an infinite number of universes will not lead to one having the requisite conditions for life if they're all identical, or only vary within set limits. Why think this is not the case? Why assume that the universes spawned by the multiverse are sufficiently random so that they exhaust all possibilities -- or at least the possibilities that entail one universe being hospitable to life?
6.2. For that matter, why assume that the multiverse spawns an infinite number of universes, or a number sufficient to make a biophilic universe possible? What if the multiverse only spawns 5,000 universes? Or 50? Or five? We have to specify a number of universes large enough to neutralize the incredibly high probabilities against a universe allowing the possibility of life, but we have no reason for assuming that a multiverse would have produced such an incredible number of universes.
7. Finally, given the multiverse, we should expect to find ourselves in a vastly different cosmos than the one in which we do, in fact, find ourselves. Roger Penrose points out in The Road to Reality that the odds of a universe having the low entropy condition that ours has is one in 1010(123). The odds of our solar system coming together by the random collision of particles is one in 1010(60) -- enormously improbable, but "utter chicken feed" in comparison to the odds against the low entropy condition being met. In other words, a universe that consisted entirely of our solar system is vastly more probable than the actual universe we have. Or, alternately, solar system universes would be much more plentiful than universes like the one in which we live, so, given the multiverse, we should expect to find ourselves in a much different, a much smaller universe.
7.1. Let me put this another way. Some of the anthropic coincidences are necessary because of the effects they produce. Universes in which those effects are met directly rather than through an anthropic coincidence are, at least in some cases, more probable. For example, when the universe sprang into existence, the property of dark energy (the stretchiness of the space-time fabric) had to be precisely what it is in order for the universe to expand at just the right speed so that gravity didn't overpower it and collapse the universe but not so fast as to prevent stars and galaxies from forming. This property has to be fine-tuned to one part in 10120. But a universe that just cuts to the chase and is created fully-formed with just one earth, one sun, and one moon would not need to meet this condition. So, all other things being equal, a smaller, simpler universe would be more likely than the universe we actually find ourselves in. Yet, superficially, such a universe would seem to be designed, moreso than ours. In fact, some people argue that if God really created the universe, we wouldn't expect it to be as expansive as it is; we should just expect the earth, sun, and moon (I think Stephen Hawking makes this point in A Brief History of Time, but I'm not sure). Such a universe, which would seem to bespeak of divine design, would be a much more likely product of a multiverse than the universe we actually have. In other words, our universe is much less plausibly explained via the multiverse hypothesis than a universe that critics of theism suggest would convince them of God's existence. This strikes me as a pretty big deal.
Now, all of this may suggest that I'm hostile to the multiverse. However, I'm only hostile to it as an alternate explanation of the universe's origin and apparent design. One of God's characteristics, at least the God of Judaism and Christianity, is that he loves to create. So it wouldn't surprise me at all to learn that there is more to reality than just two levels. To quote C. S. Lewis again:
...no man was, I suppose, ever so mad as to think that man, or all creation, filled the Divine Mind; if we are a small thing to space and time, space and time are a much smaller thing to God. It is a profound mistake to imagine that Christianity ever intended to dissipate the bewilderment and even the terror, the sense of our own nothingness, which come upon us when we think about the nature of things. It comes to intensify them. Without such sensations there is no religion.
So I have no problem with the claim that there are other universes, other realities, than our own; indeed, I would be surprised if there weren't (think of the Wood between the Worlds). Since this belief is rooted in my belief in God, however, it cannot be used to write him out of the picture. If God does not exist, I no longer have a reason for thinking there are other realities. But then the problem of the universe's origin and fine-tuning re-present themselves.
(cross-posted at Quodlibeta)