4. $\frac{\sqrt{5}-1}{2}$.

Let $F(c, 0)$ be a focus of the ellipse, which is the orthocenter of the triangle formed by the three vertices $A(a, 0)、 B(0, b)、 C(0,-b)$ of the ellipse, as shown in Figure 5. From $C F \perp A B$ we have
$$\begin{array}{l} \frac{b}{c}\left(-\frac{b}{a}\right)=-1 \\ \Rightarrow b^{2}=a c \\ \Rightarrow \frac{a^{2}-c^{2}}{a^{2}}=\frac{c}{a} \\ \Rightarrow e^{2}+e-1=0 \\ \Rightarrow e=\frac{\sqrt{5}-1}{2} . \end{array}$$