Given that deer haven’t figured out things like surgery and anti-biotics most wounds will prove fatal eventually.
Certainly I would. If it is wounded it will probably die a long painful death by infection. It it is a gut shot the probability of death is 100%.
Two years ago I had a big buck in my front yard. He “was down” but alive. He did not run even though we approached closely. I think he had been hit by a vehicle. He was put down cleanly and painlessly with a shot behind the ear. His meat was harvested and eaten.
The number one command in my lexicon is not to take the shot unless you know for damn sure it is a clean kill shot. If by error of marksmanship, it is your duty to track down the animal even if it takes you all day and kill it mercifully.
A couple of friends of mine hunt on my land. They are bow hunters and rifle hunters. As bow hunters they are extremely skilled. If not skilled they would not be allowed to hunt on my land. The greatest distance a deer has run from a bow kill on my land is 200 yards. When taken with a rifle it is behind the ear and they drop dead on the spot.
I despise non ethical hunters. They give the good guys a bad name. We do not like them!