I happen to enjoy ice skating very much, but there have been several occasions when I have passed up the opportunity due to a variation on that same fear.
The difference is, I got good at ice skating before this thought ever crossed my mind. So I don't worry about falling down. Instead I picture myself skating along at a relatively high speed, and having someone (usually a little kid) fall into my path, hand outflung. Before I can react, *schhhhhik*, no more fingers. Or hand. Or neck. (The last is extremely improbable, I know, but tell that to my imagination.)
I'm going to go be depressed for a while now from thinking about this.