If God is said to not exist because we have no evidence of his existence, then by the same reasoning and with the same certainty, wouldn't we say that extraterrestrial life does not exist?
I can see why you think that, but the reasoning is not that simple. Firstly God is not said to not exist, but presumed not to exist. Secondly when judging a claim all the evidence must be included.
Claim 1. An infinitely powerful being created the universe but there is no physical evidence of its existence.
Such a creature has to be added to all of the science we have, while not really adding any new explanations. In fact in some ways its addition is a major problem as all problems can be solved by "God did it". While that is perfectly possible, Occam's razor means that we should drop the hypothesis until such time as it usefully explains experimental results. To make matters worse religions both before and after make similar but contradictory claims cancelling out any weak evidence available from religious texts. Again this is not disproof, but does mean we need more evidence to accept the claim.
Claim 2. Earth is not the only planet to have evolved life.
We do not see anything special on Earth, except a nice combination of features (such as distance from the sun). Since there are billions of stars like ours just in our galaxy (~10%) and many stars have been found to have planets, it is likely in the whole visible universe (~100 billion galaxies) there are a truly incredible number of planets like ours. Add in the fact that life started really quickly on Earth (implying that it is not such an unlikely phenomena - though you do have the observer problem) and life sounds like it shouldn't be unique. In fact to make Earth the only life bearing planet, you would have to assume some very strange unknown feature made Earth the only one to produce life. Occam's razor then says it is actually simpler to assume that life does exist elsewhere, but is not that common or that it is mostly non-advanced (which is sensible given that for most of the time life has existed on Earth it has been single celled) to explain why we have not seen them.
So in this case, no - the two cases are different enough so the same logic cannot apply.
You see, it is not said that gods don't exist because we have no evidence of their existence. Well, ok, some people might say that. But the general idea for non-believers is that we don't believe in them because we have no evidence, not that we have reached the definitive conclusion that they don't exist. Can you see the difference? "I don't believe it exists" is not the same as "I believe it doesn't exist." But we can dig deeper than that.
The "god" claim is a supernatural one, and we have no evidence of anything supernatural, so the lack of belief is justified. Not only that, it is impossible to be proven false, it's an ill-defined concept, an unfalsifiable hypothesis. In order to believe in it, we would need good evidence, otherwise we would need to believe all sorts of contradictory things.
We do have evidence that at least 99% of the gods are made up. I mean, most of them would contradict all other gods, so by their very nature, most of the gods are definitely made up. In other words: We have evidence that there are made up gods and that people believe in made up gods.
Now, from what I can tell, people believe in gods for very bad reasons. It's basically all faith, confirmation bias, wishful thinking and having been raised a certain way. Then the so-called philosophical arguments for god's existence, none of which are the actual reasons that led anyone to believe in it, just post-hoc attempts at making convincing arguments. All they are is basically circular reasoning, begging the question and arguments from ignorance.
When we consider all of this, it does seem to make sense that gods are, actually, man-made concepts, some consider all of this evidence of god's made up nature. Evidence, not proof. See, it's not the lack of evidence of his existence that led to this conclusion, but all this other stuff.
Aliens, on the other hand... We have clear evidence that life exists in our planet, in its many different forms. So we do have evidence that life exists naturally on at least one planet. We can extrapolate that evidence and the knowledge that other planets exist, some of which may have similar characteristics to our own, and postulate that life may exist on other planets. Does that allow us to reach the conclusion that there definitively is life on other planets? No, we have no evidence of that either, but there is a good argument to be made that it is at least likely and not such a far-fetched thing in which to believe.
Q: If God is said to not exist because we have no evidence of his existence, then by the same reasoning and with the same certainty, wouldn't we say that extraterrestrial life does not exist?
These are very different concepts.
Depending on your religion, "God" might variously be omnipotent, omniscient, omnipresent, the creator of the universe, the creator of earth, the creator of life in its current form, etc. etc.
Yet every single piece of objective data ever observed can be explained without recourse to a deity. Atoms do a very good job of acting in exactly the kind of probabilistic, deterministic fashion that one would expect without supernatural control. Fossil and genetic evidence both strongly support evolution. Geology and magnetic readings clearly back up an old earth that's changed drastically through plate tectonics.
Extraterrestrial life, on the other hand, is simply the idea of life external to this planet. All the objective, deterministic evidence shows that yes, life exists, and it exists on at least one planet. Relativity also explains that travel between stars takes a very long time - even for light and radio waves. So there's a very plausible explanation, backed up by evidence, for why it's unlikely for us to have observed extraterrestrial life if it does exist.
Here is my best argument for the existence of God.
How do we prove the existence of electricity or gravity that can't be seen? Scientists can only study the effects caused by electricity or gravity because there is a scientific Principle of Cause and Effect. The Creation that includes the Universe, Life, and All There Is are Effects of a First Cause.
The First Cause can be called anything, but only the Effects of First Cause can be studied because scientists cannot study Creation from outside of the Creation. We can call electricity or gravity by any name but it does not change the effects that they cause. Since First Cause can be called anything, it can also be called God.
Crop circles throughout the planet is evidence of something that exists and maybe they can be caused by extraterrestrial life.
If it is said that God does not exist, it should be said that nothing exists. Of course the concept of the God that does not exist must be different than the concept of the God that does exist.
If it is said that God is Spirit from which all things are created, that includes the Universe and Life itself, that is a different story. Science has determined that there are billions of galaxies with billions of stars in each. Of the billions of stars, it is likely that there is life throughout the Universe with forms appropriate for the environment.
There must be intelligent life in every atom and particle that exists. Otherwise atoms would not form cells, the animal forms, plant forms, mineral forms, or matter. There is order that is evidenced by birth and rebirth of all things. Because scientists cannot study God from outside of itself, some dismiss the concept of God as described since the beginning of time. The evidence of God is the creation of the Universe and Life.
There are infinite possibilities in the creation and no material-realm logic or instruments of matter can test for all that exists. God as Spirit, is the Cause of all that exists and only the Effects that are detected in the Material Realm can be studied, including all that emanates from it.
First you must define God. The Christian definition of God includes attributes such that He is omniscient, ominipotent and ubiquitous. These are self-contradictory (anyone who wants to know why and cannot see so can post in the comments section or open a new question). On the other hand, A Taoist might say God is the universe. Hard to argue with that, but that simply is equivalent to pantheism, then.
Second, if you have defined God in a way such that His attributes are contradictory, then one can logically say that that God cannot exist. If you have defined God in the Taoist way (no pun intended), then that god can indeed exist, but that's closer to semantics.
Now, can we define extraterrestrial life? Of course: it is life that exists outside of Earth. That is unambiguous. Life on Mars, Mercury, the Sun, or on a planet 10 billion light years from Earth would all constitute extraterrestrial life. Life in space, being extraterrestrial, would also constitute that. Since the definition contains no contradiction, we must now ask the question whether there is something in the definition that seems impossible -- e.g., a creature that is carbon-based and thrives in temperatures of absolute zero is not a contradiction, but based on all present knowledge, it is impossible, since at absolute zero, there is no molecular movement, no heat produced, no energy generated. It is hard to see how such a creature could be alive or reproduce. Extraterrestrial life imposes no such constraints.
Rather, given that we exist, we know that carbon-based life is possible. Therefore, if we can establish the existence of planets similar to Earth in solar systems similar to ours (something that we have begun to do), then, at a minimum, we would have to conclude such life is possible. It would be harder to conclude it is impossible. Going further, let us say that the conditions on Earth that contribute to life occur in 1 out of 100 billion instances. There are likely more than 100 billion Earth-like planets (and we are excluding here silicon-based life; life thriving in other environments), and hence there is a very reasonable chance that other life exists.
The two propositions therefore (God and extraterrestrial life) have entirely different epistemological validities and neither one speaks to the other.
To me, James Hollomon said it best. What we have is a high probability, given our earth and its conditions, and its flora and fauna, that planets we determine through tried and true analysis to be like ours (and suns very like ours, likely with similar odds to have planets orbiting, where only a similar distance would be required to presage life,) number in large quantities across the galaxy, and the galaxies, across the universe...it's hard to argue without looking foolish that ours is the only planet ANYWHERE to contain life, is the center of creation, oh, but unlike biblical scholars of old, somehow the universe WE KNOW does not orbit the earth and, in fact in our own little solar system neighborhood, we have 100% absolute certainty the earth and other planets all circle the sun.
So is ours a sun god? Ancients thought so. We don't. We are not the center of the universe. No one thinks the sun is the master of anything beyond our planet's life - which will surely continue for a number of years...so how can there be much doubt this process repeats itself across the universe? Why not?
The centrality of the earth has been debunked, why suppose still that we are unique in the universe when all signs point to yet additional strong-and-probable likelihoods we are not?
And of god, however you perceive him/her....only of this being have we no footprint, none at all... Not one act since we've been able to wield tools that allow us to record instead of simply listening to tales...not one word we the living have heard and been able to record. No act best explained by this being...not one.
I can't see how the likelihood of extraterrestrial life and of God are even in the same ballpark.