According to Wikipedia, the Church-Turing thesis "states that a function on the natural numbers is computable by a human being ignoring resource limitations if and only if it is computable by a Turing machine."
If we made a complete simulation of the human brain, that would imply that a Turing machine can do everything that a human brain can do. Wouldn't that prove the thesis?
How would you prove that the machine is faithfully simulating a brain? How would you prove that it doesn't matter if you simulate my brain or your brain or somebody else's brain?
Church–Turing isn't something that can be proven. It's essentially just the statement that Turing machines correspond to the intuitive notion of algorithm and that just isn't a statement of mathematics. The Turing machine is the definition of computability; Church–Turing is essentially the claim that we picked the right definition, in the sense that choosing any of the reasonable alternative definitions would give exactly the same notion of computability.
Part of the issue with the idea of "proving" the Church-Turing thesis is that the Church-Turing thesis isn't a precise mathematical statement. Rather, it's the idea, or "belief" if you will, that any model of computation that could feasibly be constructed is either equal in power to a Turing machine or weaker than a Turing machine.
If we were to try to formalize the Church-Turing thesis into a rigorous mathematical statement, we'd have to somehow give definitions to terms like "model of computation" or "feasibly constructed" that are, by design, very vague. After all, the whole point of the Church-Turing thesis is to claim that working with Turing machines really captures what computation is because any other model we could have picked would end up equal in power to Turing machines. We can precisely pin down particular rules for what a model of computation is and what it means for it to be feasible to construct, but by doing so we'd limit the resulting theorem to only talking about those classes of machines. If someone later on devises some new model of computation that is wildly different than everything we've ever thought about so far and ends up being more powerful than Turing machines, then the intuitive Church-Turing thesis would be false (Turing machines don't actually capture computation!) but the rigorous Church-Turing thesis would still be true (because this new model of computation wouldn't fit into the framework developed so far).
Here's another way to see this. There are formal proofs that Turing machines can simulate computers and that computers with unbounded memory can simulate TMs. We have similar results from certain types of cellular automata (Conway's Game of Life, for example), string rewriting systems (unrestricted grammars), crystal structures (see this amazing link), etc. All of these specific cases lend support to the Church-Turing thesis, but none of these statements, in isolation, is the Church-Turing thesis.
So suppose that we prove that a TM can simulate the human brain. This would be a huge result! But that by itself doesn't prove the Church-Turing thesis because, by nature, the Church-Turing thesis is unprovable. In a sense, the answer to the more general question of "can the Church-Turing thesis be proven?" is "no," so your question falls out as a special case.
That said, it is potentially possible to disprove the Church-Turing thesis by finding some model of computation that can solve a number of problems that TMs cannot but at the same time is still something that can be feasibly constructed in the universe. In that sense, the Church-Turing thesis is more of a falsifiable scientific hypothesis than a mathematical theorem.
No, that wouldn't prove the thesis. Human beings are allowed to use machines. For example, an important consequence of the Church-Turing thesis is that computers can only compute Turing-computable functions. It is a statement about the physical universe and its capacity for computation.
No. There's no reason to believe that the human brain includes all parts of physics. In fact we're pretty sure it doesn't include many, like black hole physics and high energy particle collisions. So if black hole physics is uncomputable, that would disprove the physical Church-Turing thesis, but wouldn't prevent you from simulating a human brain.
