I would have zero problem putting a deer down in or out of season if I thought it was hovering at deaths door, I'd do it in a heartbeat. However, the deer would have to be not able to get up, or be obviously disemboweled or something.

In your case, I would not finish it off. The leg wound, mangled or not is not necessarily life threatening (you saw it, I didn't - you may decide different). Deer are amazing survivalists, and my money would be on the deer, if it can avoid infection. Once I harvested a buck with a broadhead completley through one side of the pelvis, from the year before. This year, a messed up rack buck was taken on the property that had a broadhead embedded in and through the shoulder blade from the year before. In both cases, I'm sure a couple weeks after their injury - they would have seemed on deaths door - but they both made it.

While video-taping deer coming into the corn last week (season is over - 3 weeks here) I video taped a deer come in on three legs, the hind leg on its right side was swollen the size of a melon at the knee joint. I did not have a gun with me, but considered that if I had, would I have shot it, the answer would have been no.

Life can be cruel for animals. It is difficult NOT to inject human thoughts of suffering, or pain to the deer we wholeheartedly enjoy, but we must not. Life should go on for your deer - even if its only until coyotes pull it down. If you come upon the deer at a later date, and its alive, but cannot get up, then in my opinion you have the right, maybe you are even obliged to end its life. However, as long as it can get around, I feel it would be wrong for you to illegally take the animal's life. Now if you had a tag, the decision would be a whole lot easier - wouldn't it.