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. 

[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help. 

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.

[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion [here][3].


  [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
  [3]: http://mathoverflow.net/questions/50915/a-paper-to-the-question-if-the-six-dimensional-sphere-is-a-complex-manifold