In most western states the minimum legal caliber is the .243
In the USA 43 states allow high velocity centerfires for big game hunting (this includes deer).
The remaining 7 states require the use of shotguns, and some also allow rifles that fire large caliber handgun cartridges. They do this for population density. These states are: Delaware, Illinois, Indiana, Massachusetts, New Jersey, Ohio, and Rhode Island. The state of Iowa can also be put on this list but only for certain counties.
Of the 43 states that allow high velocity rifle cartridges for big game, only 7 do NOT allow .22 centerfires. These 7 states are: Colorado, Connecticut, Iowa, Kansas, Virginia, Washington, and Wyoming. Only 3 on that list are really what we would normally call 'western' states: Colorado, Washington, and Wyoming.
The 36 states that allow .22 centerfires for big game hunting are: Alabama, Alaska, Arizona, Arkansas, California, Florida, Georgia, Hawaii, Idaho, Kentucky, Louisiana, Maine, Maryland, Michigan, Minnesota, Mississippi, Missouri, Montana, Nebraska, Nevada, New Hampshire, New Mexico, New York, North Carolina, South Carolina, North Dakota, South Dakota, Oklahoma, Oregon, Pennsylvania, Tennessee, Texas, Utah, Vermont, West Virginia, and Wisconsin.
Nationwide, most hunting kills of big game are 200 yards or less, often much less, especially for archery season.
A .223 drops a caribou or a deer, or even an elk just fine with one chest cavity hit. The .223 is helped out tremendously by the wide variety of excellent hunting bullets available. Keep the shots within 250 yards or less with a good hunting bullet, exercise good shot placement, and you will be fine.
I like the .243 because it has the energy at 300 yards that the .223 has at the muzzle. However, the .223 offers less expensive ammo and a much longer barrel life.