Balmer is currently chief executive of Moorfields Eye Hospital, a position he has held since 1992, and will take up his new post in January.
Balmer is also chairman of the European Eye Hospital Association and treasurer of the World Eye Hospital Association.
Baroness Ilora Finlay, president of the RSM, said: "Ian brings experience of enterprising developments with clinicians and an awareness of their needs. The RSM has functioned well in recent months - rather like an orchestra, however, it needs a conductor to take it on to the next adventurous pieces."
