Ratner, 42, used the gay slur during a question-and-answer session at Hollywood's Arclight cinema on Friday, and again on Howard Stern's program.
In his apology, Ratner said that it was "a dumb way of expressing myself".
"He did the right thing for the Academy and for himself," Academy President Tom Sherak said in a statement, regarding Ratner's resignation. "Words have meaning, and they have consequences. Brett is a good person, but his comments were unacceptable. We all hope this will be an opportunity to raise awareness about the harm that is caused by reckless and insensitive remarks, regardless of the intent."