In this case, it simply means "while in my care" or "on my watch". You'll hear the phrase quite a lot from both BrE and AmE speakers. In films, when someone is badly wounded, their friend or relative might be holding them and saying "Don't you dare die on me! You have to fight. You're going to be OK" (or similar).
I'm not so sure it's an idiom. To me "on me" indicates that the speaker has been receiving or hopes to receive something such as love or help from the sick person.
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