He also tweeted: "I also believe that if The Hurt Locker had been directed by a man it would not have won the Oscar for best director."
In his apology, Bret wrote in a post for the Daily Beast: "I guess I should have known better."
The novelist added: "The outcry over the Bigelow tweets was eye-opening to me in a way that nothing else has ever been. I got it. I heard it.
"I looked back at what I was doing with those tweets (quickly, unconsciously, hurriedly, drunkenly) and I have to admit they simply back-fired.
"Which is why I'm writing this. No-one asked me to write this. I simply write something like this when I'm in pain. And I've been slowly feeling a painfulness when reading all of the articles reacting to those tweets."
Bret, who has yet to hear back from the director, also said he's looking forward to watching Kathryn's new film Zero Dark Thirty, which is about the hunt for Osama bin Laden.