I think it is true that the idea that God can be active in the world is contradictory to cause and effect, or rather the methodological nature of how science is done, which assumes that all physical things are part of a connective whole and therefor have a relationship in various ways to itself, directly or indirectly. If God acts in the world, this relationship would be disrupted from the normal ways that nature is known to operate, science is contradicted in essence.
However more can be said than just this because there already are places in the universe that are known to not operate in the normal ways nature is known to. Black holes demonstrate this quite well, as does the contradiction of classical physics with quantum mechanics, according to at least the main interpretation of quantum mechanics, which is the Copenhagen interpretation as it is called. All this shows that even what is thought to be known as scientific fact is not entirely known. This means that not all natural circumstances are taken into consideration when a fact about nature is stated as unarguable fact. There is uncertainty at the extreme level of things which causes valid questions about how certain natural causal relationships really are and why and even the definition of nature itself!
For instance, if all is just cause and effect then predictability should be no problem at all. In many cases in science, this is the case, which demonstrates that cause and effect is indeed a real thing. Even here however uncertainly increases the more extreme the natural forces get which decreases predictability and therefor the view that know facts really are as certain as they are assumed to be. However an issue also arises of the exact starting point of any example of cause and effect because without this knowledge predictability is not completely accurate anyway. Then there is also the ultimate starting point issue. Did something come from nothing or has something always been infinity, and if no time before the universe or multiverse, how then does cause and effect happen anyway and so on and so forth?
If there are places in the universe where known physics breaks down, goes against common sense, or relates to philosophical questions of metaphysics and knowledge itself, or what is thought to be known about scientific fact is not quite a certain as we like, then the idea of a God working in the universe might not contradict scientific naturalism even though it appears that it must.
Our own ability to choose and make choices as well as effect the world around us accordingly, is perhaps the most personally obvious example of the issues raised here. The age old `hard problem of consciousness` rears its ugly head. If all things are simply cause and effect, then free will, choice and consciousness really don’t exist at all because the brain is just a physical object subject to cause and effect, no matter how complex it is and the processes it is subject to. Personal experience, if it is indeed real, challenges this notion!
The scientific method, which being a methodology is itself a process and therefore subject to cause and effect is predicated on what it attempts to study being also subject to cause and effect because if it isn’t, then predictably of a theory goes out of the preverbal window. As we all know, a theory can’t really be called science unless it predicts to at least some extent of accuracy. The very idea of personal choice and free will contradicts the notion of predictability. If an illusion then who or what is being fooled?
Most people would accept their own freewill even if it has limits, however they would also accept the validity of science based on causes leading to effects, and the relationships science studies to allow this process to occur. The very fact that predictability does exist in many areas of science also affirms the validity of cause and effect and of course science. This however is a contradictory position to having free will. If it is true that people do have choice and effect the world around us as a result, then God being able to do likewise could well be the same issue irrespective of God not being physical because apparently physical things only work according to cause and effect and don’t permit our own freedom of will.
Perhaps there is another way to look at it because if science and God acting in the world are mutually exclusive ideas, then so is science and human free will, and our ability to affect through freedom of choice. Are we prepared to throw out the one thing that allows for rationality to be understood, which is our own consciousness, in order to protect the idea that cause and effect and the study thereof are the universal and only epitome of rationality? It seems to me that something deeper is going on in reality that allows for real choice to be real and also the simple idea of cause and effect. Science may not be up to this study because of the way it is currently defined. Don’t get me wrong, it is good that it is defined the way it is because otherwise it would be a lot less effective which is why God should be left out of the doing of science. However its advantage may be a disadvantage in other areas of reality. This may mean that the universe works according to more than one paradigm at the same time.