When other people express remorse over things I wouldn't consider an issue, it's usually because I've already decided how I would solve the problem if it were mine. That's easy, since solving other people's problems is so much fun.
It's raw remorse I have trouble with. My kid has always had a strong interest in animals...he's had pets of every description, from birds to fish to snakes and on and on. When he was around 9 he got a King Snake that needed to be fed live mice. He never had a problem feeding it, but occasionally we'd get a feeder mouse that was too ornery for the snake to handle. Since flushing it down the toilet was not an option, my son would keep the mice as pets. Of course, they flourished under his care and soon he had sixteen aquariums full of mice in his room. He was highly attentive to them and appeared to become fascinated with their social behavior, particularly the breeding aspects. eventually he started craving more diversity than plain old white mice could provide, so I made him a deal...I'd swap him all his white mice except 1 aquarium full to be used as feeders for the snake in exchange for two 'fancy' mice of his choosing and this book about mice he would sit on the floor reading every time we went to the pet shop.
He went for it, we took 54 white mice to the pet shop and exchanged them for 2 'fancy' mice and the book, and the whole thing repeated itself, only this time in a variety of beautiful and exotic coat colors.
The female of this new pair of mice became my son's pride and joy. To me they were all just mice, but to my kid they were more like individuals with distinct personalities. Admittedly, I concede that this female was pretty cool as far as mice go. She was very docile and would sit on his desk and watch him do his homework. She went to school with him as the subject of his oral report. I swear I actually saw this mouse come to my son when he called it one time. He was very attached...
...Which is probably why our cats targeted her. One night, they succeeded in prying the top off her aquarium and made her the object of their cruel cat-games for several hours. When we found her in the morning she was alive but in bad shape, and she died later that day.
My son was wrecked. He laid on his bed and cried and cried, and I just did not know what to say or do for him. It was the most awkward position I've ever been in...it pained me to see him in such sorrow, but I couldn't feel it, couldn't find the empathy. I'm not hard-wired to handle such a flood of emotion as came out of my son that day. I only know how to shut down, and my urge was to shut him down, though that would have been much worse than doing nothing, which is pretty much what I did and always do in highly emotional situations.
If your dog gets hit by a car, going out and scraping it's remains off the pavement is the closest I can come to comforting you.