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

1. arXiv:1302.0282
Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
Xiuxiong Chen, Simon Donaldson, Song Sun
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
2. arXiv:1212.4714
Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π
Xiuxiong Chen, Simon Donaldson, Song Sun
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
3. arXiv:1211.4566
Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
Xiu-Xiong Chen, Simon Donaldson, Song Sun
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
4. arXiv:1210.7494
Kahler-Einstein metrics and stability
Xiu-Xiong Chen, Simon Donaldson, Song Sun
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)

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