Proposition 5.38 of the book Cyclotomic Fields by Lawrence Washington is as follows:
Let $m\geq 1$ be square free and assume 3 does not split completely in $\mathbb{Q}(\sqrt{-m})$. If 3 divides the class number of $\mathbb{Q}(\sqrt{3m})$ then 3 divides the class number of $\mathbb{Q}(\sqrt{-m})$.
The proof uses the 3-adic class number formula and the congruence of integer values of 3-adic L-functions to get $$\big(1-\frac{\chi\omega(3)}{3}\big)\frac{2h\log_3{\epsilon}}{\sqrt{D}}\equiv-(1-\chi(3))B_{1,\chi}\mod 3.$$ Here $\chi$ is the character for $\mathbb{Q}(\sqrt{-m})$, $\chi\omega$ is the character for $\mathbb{Q}(\sqrt{3m})$, $\epsilon,h,D$ are the fundamental unit, class number, and discriminant for $\mathbb{Q}(\sqrt{3m})$.
Then he checks that the left hand side is $h$ times something integral and therefore that if $3|h$, $3|-B_{1,\chi}=h(\mathbb{Q}(\sqrt{-m}))$. This last step uses that 3 isn't split so that $1-\chi(3)\neq 0$.
Working through the proof, I don't see where any special properties of 3 are used besides it being prime. Am I missing something, or is it true for any prime $p$ in place of 3 with the exact same proof?
You're missing the fact that the cubic unramified extension becomes a Kummer extension over the field you get by adjoining $\sqrt{-3}$, which fails to hold for primes $p > 3$.