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Telling a person who's just met you that your behavior is uncharacteristic of you doesn't hold much weight. Your behavior is all she has to go on.
I would wait until after the holidays, then call her up and ask her out as you would any other woman. Rather than telling her that the incident is out of character, demonstrate it by a normal date, and not sending another email or text until you have her reply from the previous one. Don't go into a big explanation, and stop telling her you really like her. It's too soon. Remember, she doesn't know that you stopped eating and sleeping (hopefully you didn't tell her that!) All you did that she knows is that you sent multiple emails without waiting for her reply first.
She's a med student....there are fewer busier people on earth. Make sure you can handle a relationship where communication may be infrequent.
And honestly, your reaction (not eating or sleeping) suggests that while you may be very secure in your career, and function well on your own--something about this girl is triggering something in you from your past. When a person's reaction is way out of proportion to the actual situation, it means a past hurt is being triggered. If you want to have the best chance with this girl, assuming she accepts the second date, it would be in your best interest to book a few sessions with a therapist to see why your reaction was so over the top. Here's a therapist directory to help you find someone to figure this out with: http://therapists.psychologytoday.com/rms/
Make the date by phone, if at all possible, to negate anymore overwhelming her with emails/texts. She asked you to slow down, not go away, so that leaves the door open for one more chance.