Are there NP languages whose verification is P-complete?
Every language in NP has a logspace verifier.

Indeed, let $L$ be any language in NP. Therefore, there is a polynomial time machine $T$ and a polynomial $p$ such that $x \in L$ iff some string $y$ of size at most $p(|x|)$ makes $T(x,y)$ accept.

While the witness $y$ may be hard to check, we can make help a logspace machine verify it by including the transcript of the accepting computation. We construct another machine $T'$ which accepts a new witness $z$. The witness $z$ consists of a sequence of configurations of the machine $T$, starting with an initial configuration encoding $x$ and $y$, and ending at an accepting configuration. Since each configuration has polynomial length, a logspace machine can verify that $z$ indeed consists of a valid sequence of configurations.