It's not primarily an hunting game, but you might find what you're looking for in Far Cry 3, which IS available for the Xbox360.
It's a famous, big budget game so the AI's are really good, plus it's a huge, free-roam world where you aren't pressed to follow the main storyline- just go off and do whatever you want.
The big budget makes it really visually nice too; and the Ragdoll physics are pretty much flawless, the dead animal falls when you kill it, flopping over it's front legs if it's charging, if it hit a rock it responds appropriately. very good stuff.
It's really supposed to be a shooter, so you get all sorts of weapons, so long as you can pay for them, there are LMG's, SMG's, Pistols, Sniper rifles, RPG's, Assault rifles, Shotguns. Heck, there's even bows, Flamethrowers, grenades and C4.
You can also buy attachments for a weapon, which can give it different types of scopes, silencers, extra ammo etc... Each gun can only take certain attachments, so you can't get a silenced pistol with an extra fuel tank...
You can also colour your gun, the paint job is purely cosmetic, not affecting game play- but you can colour it a camouflage colour if you want.
I should probably mention you have to look down the sight to aim; there's no little dot constantly on screen telling you where a bullet will hit.
You can get sights on some guns that have little dots on the glass parts like sights IRL have, and there are usually three lines on screen when un-zoomed, none in the centre, but indicating about where the centre area is.
One of the bigger parts of this game is the ability to throw rocks; it makes bigger or smaller sounds depending on what sort of surface you hit it on. So it'll make a big sound if you hit a tree or rock, but a smaller one if you hit soft ground.
Predators and humans will search for the sound, running towards it or slowly moving towards it, suspicious. Prey animals might run away if they feel they are in danger.
Predators will run away too- if you're fighting a pack of dogs and you kill all but a few, they'll run- or if they're hurt enough.
Gunfire alerts them.
There are some unique animals you can only get by capturing bases, they're usually significantly tougher than regular ones and can detect you far more easily- also doing more damage.
If it's raining and you zoom in with your camera or scope the rain will blur parts of the screen and run down it, making it more difficult to aim.
There are a huge range of animals on the island; Crocadiles, Sharks, Tigers, Leopards, Cassaways, Deer, Boar and many more.
On top of that there are animals you don't get any reward from, and are there for background. There might be a small rat running around, you can't tag it, loot it or anything, but you can kill it if you're fast enough. If you kill something like that it is added to your Handbook.
You can do this with birds, rays etc...
^-^ Hope this helps! ^-^