In his mémoir on what's now called the Julia set, Julia remarked (translated from French):
...in any region $D$ containing no point of $E'$ [the Julia set], the sequence of $\phi_i(z)$ [the iterates] is normal, and therefore any limit point $\zeta$ of the consequents [forward orbit] of $z$ is an analytic function of $z$ in the region $D$.
What he seems to mean is this. If $D$ is an open domain in the Fatou set, $z_0\in D$ and some sub-sequence $\phi_{k_n}(z_0)$ of the iterates converges , then the function $z\to\lim_n \phi_{k_n}(z)$ is defined and is an analytic function on $D$.
This is an interesting way of thinking about the Fatou set, but I don't see how it or any result like it follows from normality. For example, the functions $f_n(z) = z\sin(z+n)$ are normal on the unit disc, yet $(f_n(z))_n$ converges only at $z=0$.
Is there a reasonable interpretation of Julia's statement which follows from normality?
I don't know what your definition of normality is. For me, it means that there is a subsequence that convverges locally uniformly to some function (the function constant equal to $\infty$ is often accepted in that definition). By classical complex analysis result (I think it's called Weierstrass theorem?) this implies that any such limit is analytic.
It is NOT sufficient for normality to get convergence at a single point, or even to have punctual convergence.