The abelianization of $\pi_1(\operatorname{Spec}(\mathbb{Z}[\frac{1}{p}])$ is the Galois group of the maximal abelian extension of $\mathbb{Q}$ which is ramified only at $p$ (and infinity). By Class Field Theory, this field is the direct limit of the ray class fields of conductor $p^n (\infty)$, i.e., the field generated by all $p$-power roots of unity. The Galois group is thus the inverse limit of the groups $(\mathbb{Z}/p^n \mathbb{Z})^{\times}$. When $p$ is odd, this is isomorphic to $\mathbb{Z}_p \times \mu_{p-1}$ (where the second factor is cyclic of order $p-1$). So yes, this depends on $p$!
The abelianization of $\pi_1(\operatorname{Spec}(\mathbb{Z}[\frac{1}{p}])$ is the maximal abelian extension of $\mathbb{Q}$ which is ramified only at $p$ (and infinity). By Class Field Theory, this is the direct limit of the ray class fields of conductor $p^n (\infty)$, i.e., the field generated by all $p$-power roots of unity. The Galois group is thus the inverse limit of the groups $(\mathbb{Z}/p^n \mathbb{Z})^{\times}$. When $p$ is odd, this is isomorphic to $\mathbb{Z}_p \times \mu_{p-1}$ (where the second factor is cyclic of order $p-1$). So yes, this depends on $p$!