For a complex Lie group G with a real form G0⊂G, we prove that any Hamiltionian automorphism ϕ of a coadjoint orbit O0 of G0 whose connected components are simply connected, may be approximated by holomorphic O0-invariant symplectic automorphism of the corresponding coadjoint orbit of G in the sense of Carleman, provided that O is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.