Unfortunately, we have almost completely stopped hunting deer and they have become seriously overpopulated. They don't as have many natural predators left and by the way, human beings ARE a 100% natural predator of deer. We have been hunting them from the very beginning and it's not too much of a stretch to say that we evolved together. Humans have been hunting deer in the Americas for between 20,000 -30,000 years (how long have people been here? The jury is still out). Our species has hunted them continuously... pretty much lived off of them for something like 50,000 years in Eurasia and we have been going after their African kin for over 100,000 years. Deer and Humans are as linked together as Lynx and Snowshoe Hares.
Too much Bambi ... waaaay too much Bambi and, BTW, the deer are actually suffering from the lack of predation. A Mohawk friend who studies such things has noticed that the (seriously overpopulated) deer on an island in the Niagara River are actually mutating a bit because of the lack of predators presumably controlling some of the genetics. If you want to save the deer, don't stop hunting deer. Bring back cougars, bears and wolves and keep hunting deer as we always have, as long as we have been human.