A first-order sentence is valid if it is true in every possible model, i.e., if it is true for all choices of what the relation symbols, function symbols (if there are any) and constant symbols mean. A sentence is provable in some proof system if that proof system contains a proof of the sentence.
Note that provability and validity are two separate concepts, but your attempt to show that validity is recursive actually determines provability, not validity.
Validity and provability are tied together by two further notions:
- a proof system is sound if everything it can prove is valid, i.e., it only lets you prove things that are actually true;
- a proof system is complete if it can prove everything that is valid, i.e., it lets you prove all things that are true.
So your proposed method would be fine if you were using a sound and complete proof system: that would mean you could prove exactly all the valid sentences so deciding provability would be the same thing as deciding validity. Unfortunately, Gödel's famous incompleteness theorems say that there is no sound and complete proof system for first-order logic.
So, if your system is sound (it only proves true things) then it is incomplete (it doesn't prove all true things). In particular, there are some sentences $\varphi$ such that neither $\varphi$ nor $\neg\varphi$ has a proof in your system, which means that your Turing machine doesn't halt on input $\varphi$, so it doesn't actually decide any language. Alternatively, if your system is complete (it proves all true things), then it is unsound: it proves at least one false thing and, in fact, since false implies anything, it proves that every sentence is valid. In that case, the Turing machine that you thought was going to decide validity actually decides $\Sigma^*$.