Science is limited. It is like a strong dog on a chain. It rules its own area legitimately, but no more. Science is powerful and has brought us a tremendous amount of knowledge about the way in which God has set up the regularities of nature. Yet it is on a leash...
Scientists of the modern era, however, think that the dog has been let loose to roam the neighborhood. Many in our culture will claim that science is, in fact, the only way to gain knowledge. For example, Alex Rosenberg, in his book The Atheist's Guide to Reality, defines scientism thus, "....the conviction that the methods of science are the only reliable ways to secure knowledge of anything; that science's description of the world is correct in its fundamentals..." If science didn't tell it to you, you can't know it according to those represented by Rosenberg.
If I may be permitted to switch my metaphors, we may think of science as a metal detector. A young boy goes down to the beach with his metal detector and comes back with a delightful collection of soda tabs, lost earrings, and spare change. Is anyone in his family surprised that the boy found metal with his metal detector? Of course not! Notice,though, that his collection did not include sea shells, beach balls, or sand crabs. A metal detector finds what is within is capabilities.
The same is true for science. It can only find what is within its capabilities. Science goes down to the beach to study the natural world, and low and behold!, when it returns it reports to us regularities and elements of that natural world. For a scientist to proclaim that there is nothing beyond the natural world would be just like our young boy to insist that the whole world is made of metal.
Amazing how we would quaintly pat the young boy on the head with a knowing look to the other adults in the room, yet we listen to some scientists or thinkers who say the logical equivalent and nod our heads up and down in deferential agreement.
The next time you hear someone say that science is the only path to knowledge, simply pat him on the head and tell him (or her) how quaint he (or she) is.
So, can science prove that God doesn't exist? Well, is God a physical being? Not according to orthodox Christianity. And if God is not physically a part of the natural world, then he is beyond the capabilities of science and cannot be ruled out by science. Returning to the dog metaphor, science cannot investigate anything outside of its territory. It may bark threateningly, but it is restrained no matter how hard it tries not to be.
I do think, however, that God has inscribed in the regularities of nature evidence of his having designed it, and that science can detect these inscriptions. But that is a topic for another day...