For whatever its worth: $S^6$ is a ($6$-dimensional) homogenous *nearly* Kähler manifold. These were classified by [Butruille][1]. 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][2] 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remember that not too long ago there was a preprint on arXiv claiming something similar about $S^6$, but it was shortly withdrawn and I could not find any trace of it. 


  [1]: http://arxiv.org/pdf/math/0612655v1
  [2]: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.tmj/1178245052