For whatever it's 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]: https://arxiv.org/pdf/math/0612655v1.pdf
  [2]: https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-7/issue-3/Almost-Hermitian-structure-on-S6/10.2748/tmj/1178245052.full
  [3]: https://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere