What did Jesus do regarding the Jews believing that some diseases were sometimes due to sin? In relation to these two questions:

What evidence is there that the Jews thought leprosy was due to sin?

What does it mean to say that mental illness is a modern-day leprosy?

Obviously, Jesus didn't go into details about biology and psychology. Those things barely work with some people in modern society. So what did He do? Did He actually correct some people? What did Jesus say to convince them?

As far as I am able to recall, Jesus showed sympathy to the diseased by interacting or dining with them, which implies "So what if disease or their diseases are caused by sin? We still shouldn't refuse to interact with them."

That's a great lesson really, but did He also correct people? I kind of feel that He probably didn't because that wasn't really His point of being on Earth (left that to the scientists centuries later).

On the other hand, I kind of feel that such lesson or even a nudge towards things like that would have been largely helpful. I don't see any practical value in Jesus talking about calculus or gravity those don't have much to do with morality and religion, but I think some lessons or nudges about very simple biology such as some of the causes of diseases, which might eventually lead those people to think that, more often than not, diseases are caused more by physical evils than moral evils.
You ask: What did Jesus do regarding the Jews believing that some diseases were sometimes due to sin?

There's one passage in particular where explicitly corrects the disciples regarding a man born blind in John 9:


  As he passed by, he saw a man blind from birth. 2 And his disciples asked him, “Rabbi, who sinned, this man or his parents, that he was born blind?” 3 Jesus answered, "It was not that this man sinned, or his parents, but that the works of God might be displayed in him."


Admittedly, Jesus may not go as far as addressing the scientific causes of this "disease" (I'm not aware of any teaching of Jesus along those lines), and the inference to the "disease" being a result of the Fall is weak at best.  Nonetheless, he does clearly teach that a) the disease was under God's sovereignty and b) the disease was not due to the man's sin.