For whatever its worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure and then conclude that there is no complex structure on $S^6$. There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.
Finally I seem to remember that not too long ago there was a preprint on arXiv claiming something similar about $S^6$, but it was shortly withdrawn and I could not find any trace of it.