So, it's the middle of summer, your driving home and you nail a deer. You know you hit him pretty well, there's even a bit of a blood trail. Do you go home and get your bow, track him down and finish him off?
I would follow the deer, and if I can find where it dies then I take said deer, if I come upon the deer and it is still alive then how can I be sure it is a mortal wound. Sure maybe it dies in 3 days, a week, heck a month who knows...but at what point do I decide that the deer should be put down. I know what I would do, but just playing a little devils advocate. Think of rob's buck on film from last year, you take that deer after season because you injured it and assume the wound is mortal, but we all that deer lived...
Given the situation... stated in the original questions, with no other "added" variables, I would put the deer down without second thought. Really seems simple in my mind. My question would be, what happens when you check in a deer the day after the season closes? After that thought i would let the GW know that I wounded a deer, make him aware of the situation, he can come help, or he doesnt have to know it was still alive when you found him, he's already aware you were looking, that sould be enough IMO. But nothing wrong with contacting him again to tell him he's still alive, I just wouldn't call him up the day after the season and say "Hey I shot a deer last night and its still alive today, can I kill it?"
I've done It. Last year I shot my deer on the last day of the season, didn't register her till the next day. As long as It's marked legal on the tag there shouldn't be an Issue. I think a person has so many hours to get their deer registered.