Is it possible to deduce that $\mathbb C^2 \setminus \{ (0,0) \}$ is not affine directly from Hartogs's principle?

Question

It is well-known that $X=\mathbb C^2 \setminus \{ (0,0) \}$ is not an affine variety. Perhaps the simplest proof of this fact consists of showing that every regular function on $X$ extends to a regular function on $\mathbb C^2$. This is easy to do with elementary algebraic tools only. This approach is sometimes described as an algebraic version of Hartogs's principle. However, we don't actually have to invoke complex analysis. I am wondering if it is really possible to do it this way, though. Certainly Hartogs's theorem would tell me that every regular function on $X$ extends to a holomorphic function on $\mathbb C^2$. Is there some general principle that guarantees that this extension is a regular function?