More is true: the generating set $\{f_p:\text{$p$ is a prime}\}$ is dense in $\widehat{\mathbb Z}^\times$. To see this, it suffices to show that, for any finite set of primes $S$, the projection of the above generating set onto $\prod_{q\in S}\mathbb{Z}_q^\times$ is dense. Let $t$ be a positive integer whose prime factors are from $S$, and let $a\in(\mathbb{Z}/t\mathbb{Z})^\times$ be a reduced residue class modulo $t$. Then it suffices to show that there is a prime $p\not\in S$ such that $p\equiv a\pmod{t}$. Such a prime exists by Dirichlet's theorem on primes in arithmetic progressions, and we are done.

More is true: the generating set $\{f_p:\text{$p$ is a prime}\}$ is dense in $\smash{\widehat{\mathbb Z}}^\times$. To see this, it suffices to show that, for any finite set of primes $S$, the projection of the above generating set onto $\prod_{q\in S}\mathbb{Z}_q^\times$ is dense. Let $t$ be a positive integer whose prime factors are from $S$, and let $a\in(\mathbb{Z}/t\mathbb{Z})^\times$ be a reduced residue class modulo $t$. Then it suffices to show that there is a prime $p\notin S$ such that $p\equiv a\pmod{t}$. Such a prime exists by Dirichlet's theorem on primes in arithmetic progressions, and we are done.