Unfortunately, either thing is not falsifiable. That isn't saying much, though, because at such fundamental levels, not much actually is falsifiable. It's the reason that we have the Cogito.
But falsifiability only really applies to inductive justification, and even there it isn't a hard and fast rule. In most things, we tend to instead compare narratives to see which is the better fit.
Fine Tuning
We can talk about what we would expect to see under each scenario and compare that to what we observe as we explore more of nature. For instance, in the case of God existing, we would expect to see nature that reflects the attributes of God. For instance, a nature created by a God of reason with a particular interest in humans would be likely to show large amounts of consistency with regards to the laws of physics, etc., we would expect mathematical principles to be largely applicable, and we would expect that nature surrounding humanity and the subjects of the expressed concerns of that God to be unique. In other words, we would expect to find some level of fine tuning.
If, on the other hand, we were to exist in a reality without an intelligent designer, then we would expect to find a nature which is more chaotic, less directed, and less comprehensible by randomly formed minds, minds which at best, and still by chance, only had some bias, if at all, in perceiving local phenomenon. We would expect to find similar processes which created us also regularly creating other, possibly similar beings, at least in some broad sense, and we would expect much of what we find outside of our local sphere to follow inconsistent rules, and things like mathematics would need to not only be evolving, but diversifying into various conflicting disciplines in order to be useful across so many unrelated paradigms.
So, to many it would seem as if the nature we observe is more similar to what we would expect to find if there were a God than if there were no such thing. However, this does not mean that this is necessarily the case. If all things are possible, and we were to live in a world of unfiltered randomness, then necessarily there would be some bubbling up of reality which resembles the one in which we live, and it's only a matter of a Survivor's Fallacy that causes us to think that it is extraordinary.
Nature rules out certain gods
Of course, this only works for certain types of gods. Fine tuning arguments imply a creator being which values order and also which values our ability to reason. There's certainly plenty of gods people do or have believed in which would not create such a reality, or explicitly would have created something quite different.
For instance, we know that the Earth is not a flat disk on the back of a turtle, and we know that the Sun doesn't go to sleep at night in a pool of water. Likewise, gods which get drunk and fornicate with human women for fun would probably not be the sort which would create a nature such that we observe. Fine tuning arguments narrow down the attributes of a possible creator by observation of what has been created.
What is more reasonable to believe?
So, if neither is falsifiable and we cannot tell directly, are some explanations better than others? I would like to appeal to Botzmann Brains, not because I expect that it is applicable directly as I'm sure much has changed in naturalistic theories since then. However, he does pose an interesting consideration which I believe applies to any random process which requires large amounts of low probability to achieve. In such scenarios, where anything is possible over large enough samples, then we would be forced to conclude that there would be other contrary possibilities, even highly improbable ones, which would be more probable.
Consider that if somebody won the lottery twice in a row. This would be entirely possible. However, the odds would be very low, and perhaps if it happened a third time, most people would be likely to start considering other scenarios more plausible, such as a rigged lottery or the person discovering some pattern. We would start to be incredulous that this person was just that lucky. The more the odds go up, the less likely we are to seriously consider that this is all just chance.
We easily dismiss without consideration many things that are less plausible than believing nature as we observe it happened by chance entirely on the grounds of their improbability. If we do decide that we will admit such unlikely things as this as possible, then we we must likewise consider as possible, perhaps even more probable, many things we casually consider to be absurd. This includes things like Boltzmann Brains or even stranger things. So, it would seem to me that if neither case can be completely falsified, then the observation of nature can, and perhaps has, begun to reveal one possibility as the more reasonable explanation.
