Whether this is appropriate depends on the context. It implies that you treat the post of being your girlfriend as a job, so instead of breaking up with her, you fired her. In ordinary slang, we would say, "I dumped my girlfriend" if it was your idea or "I broke up with my girlfriend" if you both decided one day that you couldn't stand each other.
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AnonymousI sacked my girlfriendAnd if you are trying to say that you slept with her, it is wrong, too.