It seems that the answer is very simple (and should have been known to Uhlenbeck-Yau). Let $E$ be a stable bundle, equipped with an Hermitian-Einstein metric and connection, and $End(E)$ its authomorphism bundle, which is 
also Hermitian-Einstein, with slope 0. Then any automorphism is parallel, because a Hermitian-Einstein bundle with slope zero cannot have non-parallel sections. Then the corresponding eigenbundles are parallel with respect to the connection.
A parallel sub-bundle of a Hermitian bundle obviously splits.