There are energy recommendations for other animals. For instance, I've come across a 2000ft/lb. minimum for elk. John "Pondoro" Taylor's wrote on African rifles, and he thought a competent hunter could take on any African big game with a rifle that made 4000ft/lb.s. That's why he liked a 450/400 double rifle as an all-around gun. As far as I know there's nothing scientific about these minimums and they are just general guidelines based on observations over time.
Other people have other criteria; momentum gives more importance to bullet weight, as do Taylor's Knock Out (TKO) values, and Optimum Game Weight (OGW) calculations also takes into account bullet construction. These haven't been validated scientifically either.
The only scientific studies I've ever seen on retained energy have been on waterfowl, and were used to determine the minumum amount of retained energy a pellet had to have to penetrate the feathers and reach vitals.
What's good for the goose is good for the, well, other animals, too. Energy alone doesn't kill; the projectile has be able to hold together and reach the vitals on the largest animal you may encounter from whatever angle you may have to shoot it from.
There are .22 caliber big game bullets (Nosler Partition, Barnes TSX, etc.) but most .22 bullets are not designed to hold together on large animals. Most .22 caliber rifles have twist rates of around 1 in 12" designed for optimal performance with the lighter, frangible varmint bullets. The energy may be just barely there, but if you use the wrong bullets or your rifle can't shoot the right bullets then it's still irresponsible. If you have a rifle that can shoot big-game bullets, or rebarrel to a better twist rate, and the deer are small then it'll work under ideal conditions if you're a good shot and a patient hunter. But it's marginal and, in my opinion, doesn't show the proper respect to the animal you're hunting.