I have hit a dog and a cat (thankfully no people but came very close once to hitting a child). While I don't like injuring animals, animals are not people. If I have the means to avoid them, I will. But if I don't because of traffic or other conditions, I just have to hit them and live with the consequences.
That happened once with a cat that ran into the street and under my tires. I could see the cat in my rear view mirror flopping around in pain but I could tell it wasn't going to live long. When I got to the stoplight, some old woman started honking her horn and cursing at me as if I had chosen to murder some innocent kitty cat.
Motor vehicles rule the road. Nothing has priority over a motor vehicle on the road. If you don't think that way, you'll never go faster than 20 mph.
Maybe the old woman was quite reasonably thinking that it was pretty inconsiderate of you not to stop and get the cat out of the street and see if it had an ID tag with phone number. Sometimes you can't help hitting an animal, and can't save the animal. But you may well be able to minimize the grief of its owners, and of children who might find their beloved pet mangled in the road. The pain of not knowing what happened to a missing pet, and the efforts many people go to in trying to locate a lost pet, are considerable. Just a couple of minutes of your time might have alleviated some of that. Hopefully someone else, like the old woman, did the task for you.