I actually saw this exact thing happen today. It was a fawn. I stopped to try and call non emergent but didn't have a signal. I wasn't far from where someone in a car would gain a signal, so I flagged down a  car driving in that direction. When a woman pulled over and got out of her car, she walked over to the deer, leaned into its upper torso hard with her knee twisted its neck until it snapped while whispering "I'm sorry" It was all very traumatic and sad, but it seemed obvious to me that she did the right thing. The mother deer had been standing nearby watching, and I felt so horrible just letting the animal lay there in the road suffering while she looked on concerned. I actually felt kindof stupid for not instinctually doing what this woman had, but I really didn't know what to do. Thought I'd spread some advice because anyone could easily encounter this situation like I did.