A precise statement and proof of the relationship between stability and the existence of Calabi-Yau metrics is in:

1. arXiv:1302.0282, Xiuxiong Chen, Simon Donaldson, Song Sun, 
*Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof*

2. arXiv:1212.4714, Xiuxiong Chen, Simon Donaldson, Song Sun,
*Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π*

3. arXiv:1211.4566, Xiu-Xiong Chen, Simon Donaldson, Song Sun, 
*Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities*

4. arXiv:1210.7494, Xiu-Xiong Chen, Simon Donaldson, Song Sun
*Kahler-Einstein metrics and stability*

5. arXiv:1211.4669, Gang Tian, *K-stability and Kähler-Einstein metrics*

(so I understand; I haven't read all of this)