We can use the Loop Theorem to show that if $\Sigma$ is a minimal-genus Seifert surface for a link $L$, then $\pi_1(\Sigma)$ injects into the knot group $\pi_1(S^3 \setminus L)$. An orientable connected surface with nonempty boundary and abelian fundamental group must be a disk or an annulus. Therefore $\pi_1(S^3 \setminus L)$ is abelian only if it is $\mathbb{Z}$ or $\mathbb{Z}\oplus \mathbb{Z}$, implying that $L$ is an unknot or a (nonsplit) two-component link with genus zero.
If $\,\pi_1(S^3 \setminus L) \cong \mathbb{Z} \oplus \mathbb{Z}$, how do we know that $L$ is the Hopf link?
I know that this was proven by Neuwirth in his paper "A note on torus knots and links determined by their groups" (1961). I can't access that paper, and I would be satisfied with an overview of his proof (or another reference). But it would be great to know if there have been any simple arguments discovered since then.
Edit: See my first answer below for a summary of Neuwirth's proof and my second answer for another argument and a proof that $\pi_1(\Sigma)$ injects into $\pi_1(S^3 \setminus L)$.