Can one use the PCP theorem to prove correctness of deternimistic algorithms?
The statement you state is not the PCP theorem. The PCP theorem is a characterization of NP. 

The statement you mention is much easier to prove. It says in effect that a polytime algorithm that uses $O(\log n)$ random bits can be made deterministic while keeping it polytime (given reasonable completeness and soundness properties). The proof is very simple – we can run the algorithm over the $2^{O(\log n)} = n^{O(1)}$ many choices of random bits, check how many times the algorithm accepts, and so determine whether the input belongs to the language.

At most you can think of this as a derandomization procedure – if your algorithm uses only logarithmically many random bits, then you can make it deterministic by "exhaustive search".