There is a proof in section 2.2 of 
<a href="http://books.google.com/books/about/The_Geometry_of_Four_Manifolds.html?id=LbHmMtrebi4C">The geometry of four-manifolds</a>, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.

(While it's possible to prove this integrability theorem by quoting Newlander-Nirenberg, the PDE problem that underlies it is considerably easier to solve than the N-N problem, because the bundle is decoupled from the coordinates on the base.)