This is related to Are there statements that are undecidable but not provably undecidable and the links referred to there which addresses analagous questions, showing yes there are such statements.
This is an erroneous armchair argument to the contrary and I am wondering what is wrong with this line of reasoning. I have thought about it myself and asked others, but we aren't clear what's wrong. For a proposition P consider the statements
1) P can be proven.
2) P can be disproven.
3) Neither 1) or 2) can be proven.
Suppose 1), 2), and 3) cannot be proven. Then 3) is true.
Case 1: If we accept this as a proof of 3) we get a contradiction. So either 1), 2), or 3) can be proven.
Case 2: If we do not accept this as a proof of 3). Note 3) cannot be disproven or else 1) or 2) would be provable. So 3) must be undecidable. So 3) can be either true or false without contradiction since unprovability means independence. But this is absurd since if 3) were false 1) or 2) could be proven. So either 1), 2), or 3) was provable.
First of all, what you have written certainly is not a proof of 3). It is a proof of the statement "if 1), 2), and 3) cannot be proven, then 3)". So, if you can prove that 1), 2), and 3) cannot be proven, you would have a proof of 3). But it may be that 1), 2), and 3) cannot be proven, but you cannot prove that fact.
In your "Case 2", there are also errors. If 3) is disprovable, that does not mean that 1) or 2) is provable. It just means that it is provable that 1) or 2) is provable, but you cannot deduce the truth of a statement from its provability. (If you actually have a proof of a statement then you can conclude it is true, but if you just abstractly know it is provable as we do in this argument, you cannot conclude it is true.)
Moreover, even if 3) is not disprovable, there is no contradiction. Here you essentially just repeated your first error: you have shown that if 1) and 2) are not provable then 3) cannot be false. That does not prove 3) actually is true unless you first prove that 1) and 2) are not provable.