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Yes I do believe it is impossible for a predator to kill off a healthy deer herd.
Nature has a way of balancing things out,if the deer population becomes large ,the predator population also becomes larger,population of deer shrinks,predator population also shrinks.Been that way for what,probably 10's of thousands of years?
Man is the only animal on the planet that kills for sport, some species kill other species randomly, but it is usually to eliminate competition.
Man is the only animal that I can think of that has caused other animal species to go extinct,or near extinct.
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The "balance" of nature about which you speak is somewhat fictional. Nature's balance is really continued imbalance. Contrary to our perceptions, stagnate populations are detrimental because they don't adjust to the environment which is never the same from year to year Everything stays in flux. Those living in the SW may still be familiar with the population explosions of jackrabbits every few years which are followed by coyote population explosions soon thereafter. These are what many of the ranchers notice...first the jacks that compete with their cattle and then the coyotes that kill their calves. This noted pattern is probably because of a lack of predators and other food as influenced by humans, but in the boreal north, the snowshoe hare's population cycle occurs and has been tied to the lynx, though the tie is apparently observational bias of sorts.
http://www.pnas.org/content/94/10/5147.full
While you may think it is impossible for a predator to kill off a healthy deer herd, you aren't looking at the bigger picture. It just isn't the deer herd that feeds a given predator. The predator (and there is almost never just one type) may decimate a local population down below healthy numbers. The population may rebound, but not necessarily via the local "herd," but from encroachment from other herds.
If you actually mean a specific group of deer as a "herd" then yes, a predator can kill off a healthy herd of deer, but deer herds are quite flexible and disperse and combine with other individuals such that a herd may not remain homogenous for any extended period of time.
Humans are the only animals that kill for sport? Well, they are the only ones that call is "sport" when they kill for reasons other than food or other needs. James Froude proclaimed that wild animals never kill for sport back in the 1800s. He was a historian and novelist, but not a biologist. Motivation is not always clear cut. Certainly there are animals that kill for reason other than food, protection, sexual dominance, or territorial dominance. Dolphins will kill porpoises and not eat them and porpoises generally don't necessarily pose a risk for dolphins. The killing isn't well understood, but has been observed repeatedly. Some have claimed it is for sport. It does not appear to be for protection or territory.
Various species of cat are known to play with smaller game before killing it.
http://www.africanwildlifeguide.com/...g-for-survival So it could be argued that the "playing with" of the game is for sport. Weasels do as well. Foxes, possibly so.
http://www.askabiologist.org.uk/answ...ic.php?id=4741