The fact that a stable manifold is diffeomorphic to a (real) Euclidean space is a consequence of Thethe Stable Manifold Theorem, see for instance [Katok&Hasselblatt, Introduction to the modern theory of dynamical systems. chap 6 sec 4]:
By the Stable Manifold Theorem for every $p\in M$, there is a local stable manifold $W^s_{loc}(p)$ which is diffeomorphic to aan Euclidean ball, and the global stable manifold satisfies $W^s(p)=\bigcup_{n\geq 0} f^{-n}( W^s_{loc}(f^n(p)))$. Thus $W^s(p)$ is a monotone union of Euclidean balls, hence $W^s(p)$ itself is diffeomorphic to an Euclidean ball. Here is nothing to do with holomorphic map.
To determine the complex structure of a stable manifold when we have a holomorphic diffeomorphism, is more difficult. When $W^s(p)$ has dimension 1, it can be shown that $W^s(p)$ is biholomorphic to $\mathbb{C}$ but not $\mathbb{D}$, see for instance [Bedford&Smillie, Polynomial diffeomorphisms of $\mathbb{C}^2$: currents, equilibrium measure and hyperbolicity. Thm 5.4]. It is conjectured that if $f:M\to M$ is a holomorphic diffeomorphism with aan invariant hyperbolic set $\Lambda$, then a $k-$dimensional stable manifold of $p\in\Lambda$ is biholomorphic to $\mathbb{C}^k$. As far as I known this is still open.