I'm a runner. Did a 12k last May and after the race, my second toe on my left foot felt weird. Kinda numb and it was slighty darker than it's neighbors. 4 days later the whole came off in it's entirety. Zero pain and was very strange. Months later it came back perfect and have no issues. I had a pedicure before I ran the Phx PF Changs half marathon and my toes were fine. I've seen stickers that say "Toenails are for sissies". Lots of my runner firends have tons of stories of losing nails at various times.
Yours was, of course, WAY WORSE! OUCH!!
And I don't think you need a Dr. Keep it very clean and apply Neosporin and let it air out. In a week you'll feel a ton better.
Keep us up to date! Best wishes, bro!