Yes, it does. This is sometimes called Kuran's theorem, proved by Ülkü Kuran in 1972, see [1], after preliminary works by a number of other researchers.
Reference:
[1] Ülkü Kuran, On the mean-value property of harmonic functions. Bull. London Math. Soc. 4 (1972): 311–312, DOI:10.1112/blms/4.3.311.