Free right on green light with pedestrian crossing?

I live in Florida. Here's a situation which confused me this morning. I am at a signal which is green. I am taking a right, when I see a pedestrian (a cyclist) crossing right in front of me. I slow down and almost stop, but then realize if I do stop, the car behind me is going to rear end me, So I continue going, and the cyclist stares at me. Who has the right of way in this case ? If it was a red signal, he has the right of way with no doubt. And in my case, if I do have the right of way - if he still decides to cross and I hit him, is that my fault ?