"I made a series of remarks that were meant only to be funny, but in the cold light of day, they are in fact thoughtless and insensitive - and not funny at all. To anybody I've offended, I'm profoundly and deeply sorry, and I regret and apologise for any injury they might have caused."
Rudin, who produced The Social Network and The Queen, made the emailed comments in what appear to be a series of angry exchanges with Amy Pascal, co-chairman of Sony Pictures Entertainment.
The studio has been targeted by hackers with suspected links to North Korea, who have leaked large amounts of information from Sony's computers.
According to the messages Rudin and Pascal began swearing at each other because Jolie wanted director David Fincher to oversee her planned remake of Cleopatra instead of a Steve Jobs biopic written by Aaron Sorkin.
Rudin, pushing the Jobs movie, wrote to Pascal: "There is no movie of Cleopatra to be made (and how that is a bad thing given the insanity and rampaging spoiled ego of this woman and the cost of the movie is beyond me) ... I have zero appetite for the indulgence of spoiled brats and I will tell her this myself if you don't."
He predicted Cleopatra would be a "$180m ego bath and giantbomb" and "career-defining debacle for us both". Jolie has previously hinted that Cleopatra may be her last acting role before she becomes a full-time director.
The actor has not responded.
Variety, the film industry magazine, called the emails "ugly" and did not question their authenticity.
Suspicion over the hacking has centred on North Korea, which has expressed anger over Sony's forthcoming release of The Interview, a comedy about an attempt to assassinate its leader, Kim Jong-un.
- Daily Telegraph, AP