Making logical proofs that something is impossible is difficult. In particular, you have to define the words you are using carefully enough to be able to use logical terms like "true" or "false." When discussing miracles, this can be difficult to do.
Usually, when faced with such a question, I ask people to define "miracle." You were kind enough to already do that. So, instead, I'd ask you to define "supernatural." For instance, dualism declares that there is a metaphysical mind which interacts with the body. Is that metaphysical mind "supernatural?" If so, then a very popular class of philosophy not only declares such interaction to be possible, but declares that it is happening continuously, every moment of the day.
If the human mind is excluded from the supernatural (along the lines of "well, we're natural beings, so even though metaphysical mind was involved, it's still 'natural,' or through claiming physicalism rather than dualism) then we have to start discussing what is left that can be supernatural. Now we're explicitly excluding the human mind, so we're going to have to explain how we define supernatural more strictly. Otherwise we do see a possible argument that miracles are logically impossible because we may start to redefine "supernatural" to become "things that don't interact with the natural world." Once they interact, they become natural.
There are other definitions we can explore, hidden in the question, which do not require one to hold dualist views to appreciate. For example, it is not uncommon to see the belief that the universe is non-deterministic: that there is some randomness to it. This randomness appears so deeply in our worldview that it can be difficult to argue it isn't "real." Several of the most popular interpretations of quantum physics include a concept of a random variable when discussing collapse. Chaotic systems can be mathematically shown to be arbitrarily sensitive to their initial state, making it very hard to observe whether the laws of physics are being obeyed or broken deep in their core. So far, science has found no reason to believe the laws of physics are anything but obeyed in these circumstances, but it turns out to be terribly hard to make a "logical" proof thereof because we simply cannot know enough about the system to do so. However, these random events are unique, and thus it would be impossible for us to prove that a supernatural event did not occur. A supernatural intervention would be indistinguishable from a tremendously "lucky" event.
And then, of course, there is the big bang. All causality models get tremendously murky around this event. Perhaps it's a miracle? I'd say that depends mostly on how you end up defining it. Were there natural laws beforehand? Some cosmologists say yes, some say no. Nothing is quite definitive yet.
In all, I think the answer is that there is no proof that miracles are impossible. In fact, depending on your preferred definition of many words, you may actually be able to make a meta-proof: a proof that it is impossible to logically prove miracles are impossible! The only way I know of to make a logical proof thereof is to intentionally choose your definitions such that you axiomatically declare miracles to be impossible, and thus axiomatically "prove" it.