Keeping the analogy within reason, one should approach hunting with a handgun (especially such a minimum as a 4" 357) as you would if you were hunting with a bow. Yes, I know about all the amazing pistol assassins of antlered game that make 200 yard shots that a mere mortal such as myself that has been hunting for 40 years wouldn't attempt, but that's my opinion.
Get close, get closer if you can. Use a rest - shooting sticks or a cane type brace. Use as much gun as you can realistically control and shoot WELL. PRACTICE shooting from field positions. Learn deer anatomy and only make high percentage well placed shots. And use HEARING protection. Amazing how many first time handgun hunters learn that one too late . . . .
High powered rifle bullets for the most part incapacitate/kill by shock to a great degree (think of the flesh as wet sponge that the bullet treats as if driving the liquid away from and out of the impact area); pistol bullets actually are more akin to arrow impacts in the way they kill - tear blood vessels through a vital area and try to penetrate with a pass through shot to let air in and blood out. High speed hollow points are not always the best choice. Old Elmer had it right using as big a solid bullet as possible with a blunt edge that impacts a wallop while driving through and leaving a large hole.
Deer can certainly be easily taken with a 4" 357. In the right hands and right conditions they can be taken with a .22 or even a high speed pellet gun. But none of those are quick killing choices without precision placement and maximum damage achieved from the shot. Under excitement (duress) and field conditions it's best to plan on having a little "overkill" at your disposal for when things don't go exactly right. There is a component of shooting a wild animal with a handgun that many (usually who have never done such) overlook - when a man receives a wound from such his mind tells him "I've been shot !" and depending on how "hard" and where the wound is, his mind will tell him "I'm might die!" and shock begins to set in from the mental as well as physical aspects. A wild thing such as a deer on the other hand, only reacts with fight or flight reaction from being startled and hurt and doesn't know that when you shoot it it is supposed to lay down and die. Even a mortally wounded deer with only seconds or minutes to live can cover a LOT of territory QUICKLY and "disappear" more easily than you can imagine if you've never had to track one. The idea then is to get the job done as quickly and efficiently as possible. A 357 hollow point slug may well be fatal in a shot not immediately incapacitating - but you may never see the deer again, either.
But there's nothing more satisfying for an enthusiastic handgunner than to bring home venison with one of his favorite tools. I've just seen too many novice bow and handgun hunters wound and lose too many deer. Best to start out with the right tools, attitude, and skills as much as possible.