Fundamentally, theology and science cover two different areas of reality. Science ultimately explains the “how” things occur and theology is more about the “why”. This can be a confusing thought at first as it makes sense that if God created the universe, science should see evidence of “how” He did it. And we do see evidence of “how” the universe was created, from rates of expansion to background radiation, but none of this evidence seems to point directly to God. Atheists might say that this is a concern for Christians, but it really shouldn’t be.
Keep in mind that we are talking about an entity that isn’t subject to the laws of our universe, He created them. It seems reasonable to assume that if God created the system of constants that govern our universe, then He is not subject to them. If He is not subject to them, then why should they provide direct evidence that He exists? The only evidence that we should expect to see in a created universe are limitations. Limitations to speed, time, temperature, distance, age, types of particles, and more.
Do we see such limitations? In many cases yes. In terms of speed, matter is theorized to be limited to slower than the speed of light and nothing has been confirmed to exist that is faster than the speed of light. Time is very debated, but does appear to be related to the speed of light as well. Temperature has a theorized absolute minimum and maximum. The universe has a theorized age, meaning that it is not infinite and has a beginning. Distance is an interesting one. The universe is not infinite in distance, but has no end. The are only so many galaxies but there is no “end” or “edge” galaxy. Supposedly, no matter where you are in the universe it will always appear as though you are in the center, yet it is not infinite. There are many questions about how this works conceptually, but mathematically it presumably is sound. The math, however, leads to the possibility of something existing outside of our universe. Something that is not required by our universe, but could mathematically exist. This has lead to speculation about multiple universes, or the “multiverse”. There are many variations of this theory, but essentially there is a scientifically admitted possibility that other universes exist which cannot be seen from our own. While I don’t want to get too deep into this subject, I would like to say that Sheol (the afterlife) could simply be another universe.
At any rate, we do see limitations in our universe. We can ascribe these limitations to mere chance or to a creator. Either way, we are making a choice based on faith. Faith that there is something more, or faith that there is not. The interesting thing about chance is that anything can be said to have occurred by chance, even an intentional act.
Now, back to the original question. Could science ever prove that God does not exist? Not really. The day that science has explained how everything in the material world works and how it formed, there will still be a fundamental piece missing; why? Why is there anything at all? Science doesn’t ask why we are here, it asks how we got here. If God spoke and mankind was created as a result, would we find evidence of God’s hands at work? Probably not, especially if God spoke things into existence as the Bible says, and we still wouldn’t know why any of it exists. We may eventually understand how everything in the material world works, how it was physically created, but we still won’t know why. That is for philosophy and theology to tackle.