For a complex Lie group $G$ with a real form $G_{0}\subset G$, we prove that any Hamiltionian automorphism $\phi$ of a coadjoint orbit $\mathcal{O}_{0}$ of $G_{0}$ whose connected components are simply connected, may be approximated by holomorphic $O_{0}$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal{O}$ is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.[-]