I am in tears reading this. I had a similar experience. I worked at a day care and we did van runs to local schools to pick up children. This one day we had to take two vans. I was in the front van and my co-worker was following me. Me and this other person didn't get along too well.
On the way a black cat came out of no where and I didn't have time to do anything. I cried so hard on the way to the school. When we got there the other driver just came up to me and grabbed me in the biggest hug. She helped me calm down, because I didn't want the children to see me so upset and ask what happened.
On the way back to the center we passed the spot and the kitty was laying lifeless in the middle of the road. I couldn't even do anything about it. I tried my hardest to control the tears from flowing again since I had the kids in the van.
I can only take comfort that it didn't appear that the kitty suffered too much.
Please don't beat yourself up about it. You did stop, something I couldn't do.