Why can't science prove or disprove the supernatural? Because by definition, science deals with the natural world. It does not have the tools nor the ability to deal with the supernatural world, assuming one really exists. David Eller explains:
However, when religious entities really are outside the scope of nature (supernatural or transnatural), they are also undetectable. But what can we possibly do with the undetectable? How can we study it? How can we know about it--or even know if it exists? We cannot. It is a null category as far as human knowledge is concerned. Science, therefore, limits itself to the detectable because that is the only thing we as humans have access to and can know (Atheism Advanced,p. 211). This is why movements like the Intelligent Design movement are not really science. ID is a philosophical inference based on data drawn from science but it is not science. Eller continues:
But, a religionist can respond, gods or spirits or forces are detectable, in the design of nature, or the curing of illness, or the manifestation in oracles and divination, etc. The problem with such detections is that there are other possible interpretations of them. Some illnesses respond to medical treatment, and sometimes the body fights them off. In other words, there are natural explanations too. The "design" of nature can come about through natural processes, like natural selection--and few religionists detect in the organization of a hurricane a divine hand at work (and certainly not a benevolent divine hand) (pp. 211-12).Furthermore, supernatural explanations are typically inserted where there is not yet a natural explanation ("god of the gaps"). Man is impatient. He wants an explanation and he wants it now. Religion provides a quick and easy answer for the unexplainable. The problem for religionists is that their supernatural explanations have been shown time and time again to be premature because a natural explanation later surfaced. What is a believer to do at that stage? Well, they retreat to the idea that their god is actually working through the "natural" means. For example, with regard to evolution, you have the theistic evolutionists who say that God uses natural processes to bring about the universe. How can one disprove that? He can't. Its a fool proof argument that provides the believer with a safe house for his faith. The question, of course becomes, as even the fundamentalist argues: "Why does anyone need a god if it all can be explained through natural processes"? This is why fundamentalists are so adamant in their rejection of evolution. If the ultimate mystery of how we got here can be explained without a god, then there is nothing left for a god to explain. They sense the cruciality of this issue and have drawn a line in the sand (see "Why We Fight About This" by Peter Enns).
The reason that science cannot employ supernatural explanations is because it would mean an end to science. The scientific method is based on observability and repeatability, neither of which can be said about supernatural forces. The essential element in all religions is that there is a non-human personal agency at work behind the scenes causing various phenomena in the physical world. Once again I turn to Eller for further elaboration:
Science rejects or dismisses the element of agency in nature because agency makes science impossible and paralyzes all human knowledge. It has this unavoidable effect for two critical reasons. First, the entire project of science depends on the regularity and predictability of nature, and agency makes events irregular and unpredictable. By definition, agency or will is not completely determined by pre-existing conditions; it establishes a certain zone of freedom for agents. They have their own desires or interests or wills independent of conditions. Therefore, we never know quite what they will do. The exact same situation can lead to completely opposite results if the agents so choose; there is no connection between cause and effects. This frustrates and precludes the possibility of ever knowing with any degree of confidence what will happen next(p. 214).
To summarize: Religion functions on the personal premise, that some or all natural events are the results of the reasons of agents. Science functions on the impersonal premise, that all natural events are the effects of antecedent and non-agentive causes (p. 215).
Christians will often argue that science is just biased against the supernatural and has ruled out God a priori . The truth is that science has to rule out supernatural explanations because it is based on observable phenomena and the supernatural cannot be detected or observed; and, furthermore, if personal agency is the cause of natural events, it is totally up to the will of that supernatural person as to when and where he will intervene, and there can be no repeatability or controlled conditions which is essential for science. Finally, if one accepts a supernatural explanation, then the research ends, science has no more work to do on that problem. It would mean the end of any advancement in knowledge.