Direct proof that a parallel almost complex structure is integrable

Question

Let $(M,g,J)$ be an almost Hermitian manifold nad $\nabla$ be the Levi-Civita connection on $M$. If $\nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which implies that $J$ is integrable. But the condition $\nabla J=0$ is stronger than $N_J=0$, so my question is: Is there a direct proof of this statement without invoking the Newlander-Nierenberg theorem?