Many scientific disciplines are such that experiments are impossible. The effects of childhood abuse, for example. Scientists who study such matters are generally very sensitive to the fact that the empirical work they do cannot prove causation in the way that controlled studies can.
What a well-designed correlative study can prove or disprove is that of a long list of proposed explanations for some effect, one is the most likely. For example, you can make a study that determines which of the claims "A causes C" or "B causes C" is more likely. If you do enough of these, against every conceivable other thing that might effect C, then you can reasonably assert that you have evidence that A causes C, but you have to remember that it is a different kind of evidence from in-lab controlled experiments. (Sometimes it is better evidence: lab environments can be poor approximations of the actual world.) ((Yet another kind of evidence would be a proposed mechanism. If you can tell a convincing yarn about why A causes C, which builds on a series of well-established causal relationships, then you may claim you have evidence for your causal assertion. I think that this kind of evidence is generally the weakest, but it is often the one that people like the best, since most people understand the world through stories.))
But it is important to remember that just the existence of a correlation need not have much to do with a causal relationship. Here's one that comes to mind. It is a fact that a history of childhood victimization predicts for (is correlated with) lower adulthood weight (the effect is weak, but statistically significant). However, the best guess for the relationship between childhood victimization and weight is that victimization causes the victim to be overweight in adulthood. This result shows up in the empirical data after you "hold other variables constant". In particular, childhood victimization strongly correlates with tobacco use, and tobacco is known to make people thinner. But if you child-abuse-victimization with weight within either the tobacco-using population or the tobacco-non-using population, you do see a correlation between abuse and obesity. Again, it is always possible that there are other better explanations for this effect, and to test them you go out in the field and measure all your proposed variables. My understanding of the result "child abuse causes obesity" is that it is more likely than any other explanation that people have thought of, in the sense that every variable scientists have thought of to hold constant or covary or all the other things they can do in the statistical models seems to lead to the asserted conclusion.