Spanning set for $L^2(S^2)$?

Assuming you mean that their span (finite linear combinations of the $f_v$) is dense in $H$ (i.e. any element can be written as an infinite linear combination of the $f_v$), this will be true.

One way to see this is to see that the span of the $f_v$ is dense in the space of continuous functions on the sphere w.r.t. to the uniform norm $$\|f\|_{\infty}=\sup_{x\in \mathbb{S}^2}|f(x)|$$ (this follows from Stone-Weierstrass).

Then, you can use that the continuous functions are dense in $H$ and that $\|f\|_{L^2}\leq \|f\|_{\infty} Vol(\mathbb{S}^2)$ to get that the span of the $f_v$ must be dense in $H$.