I understand your fear; but you should not be afraid. What happened to you happens to many people (myself included) In my case, it was avocado.
Any way, not nice but a piece of mango skin is not meant to give you major problems if you survive the first few hours without a need to run to the hospital.
Your stomach may/may not digest the piece. Whatever it does, you will be fine.
You may have mint tea without Sugar if you want to soothe your stomach...or simply keep yourself fasting. I believe in fasting myself, and this experience, as you say, may be a chance to do an extended fast now.
Stay away from the medical doctors if you are not having a real emergency (they are good for emergency care and for diagnostic only). They are not good for other things. You do not need people to increase your fear.
A good decision is also, if you break your fast, is to drink pineapple juice, as the previous poster advised to you. And those enzymes.
Really, at this point, if you can be quiet, you will feel better and better and better....
I guess that for you-respectfully said- your next BM will be reason for a happy party.