In my continued searches for modern proofs of Newlander-Nirenberg, I found this great source: it's "Applications of Partial Differential Equations to Some Problems in Geometry", a set of lecture notes by Jerry Kazdan, which are available on his website at UPenn. The proof of Newlander-Nirenberg in here is based on Malgrange's proof, which is also in Nirenberg's book (mentioned by Morris KaLka in his answer above), but these notes by Kazdan cover a lot of basic geometric analysis theorems, so they're an excellent resource. I wish I knew about these when I was in graduate school...