Imagine how that animal would have suffered if you hadn't finished it. As hunters it is our job to make as quick and humane a kill as possible. Even though that doesn't always happen, it shows responsibility and respect for the animal to follow up in any way necessary. I rifle-shot a 7 point buck five years ago that dropped instantly in some heavy weeds. I got to it and realized that the deer was not done yet, we both knew what was going to happen next and he let out a bawl that will always be in my soul. I do my best to make every shot count so I never hear that or put an animal in the position to make that sound again. This is going to sound strange, but if I was being hunted, I would want to be dispatched by someone who was going to do it the most efficient way possible. To be there one second and gone the next.
I know a man who saw a big buck get struck by a vehicle. The driver left the scene and this man got a light out of his car and found the buck, which was struggling to get away on broken limbs & other injuries. He called the police who said they'd have someone there "in a few minutes". Forty minutes later, this guy finished the deer with a handgun that he kept in the trunk because he couldn't stand to see & hear the buck's struggles. Of course, the Law arrived about the time the sound of the shot quit echoing and the officer started giving the man grief about what he'd done. The man told the officer that he couldn't watch that animal suffer any longer and if there were any fines or fees, he'd gladly pay them just to end the ordeal. He waited for a Highway Patrol officer to give him a permit to possess a road-killed deer. Years later the buck's rack was scored and it would have made the B & C book.
Compassion is not a bad trait to possess. Good for you for caring. Deer are more than moving targets. I watch & feed deer daily and have known some since they were a couple of weeks old. They each have different personalities. Some deer are my friends and I would never hunt anywhere near my house because of that. It would be like hunting someone's pet dog. There is a doe here I call Sally who has been coming here for 7 years since she was in spots. It makes my day to see her & have her take food from my hand. Since I take from somewhere, I give back somewhere too. The fact is that their numbers have to be controlled and it doesn't hurt that they are high protien, low fat, and tasty.
At some point in your hunting career you will have to deal with some unpleasant situations which you may or may not be responsible for. That's part of being at the top of the food chain and accepting the role of hunter. Some people will step away, some will not. How you handle it is up to you.
I'll attach a picture of a little buck I named Sonny, who went his own way shortly after he polished his first set of antlers. He was also a daily visitor and would only come to me and avoided my wife & all others for whatever reason. He was 3-4 weeks old at the time of the picture. I hope he's still out there and has gotten to maturity.