Depends on where you live, I suppose.
Here in Far Northern Canada it is highly illegal and for good reason. The deer are MUCH larger here than the Southern United States.
Shots are also generally taken over great, open expanses between 200-350 yards in my experience. Shots are usually taken with a tall bipod while sitting on your foot.
Flatter shooter calibers such as the 270win and 300 win mag are preferred. Less flat shooting calibers are good out to 350 yards too such as the 30'06 and 7mm'08 or the 338wm. The bullet drop is greater on these but a bit of practice to know your drops and you should be okay.
A 223 Remington is a comical joke for this application, particularly since you're just as likely to encounter a giant Moose or Elk or brown bear.
Even if it were legal here, I wouldn't do it for ethical reasons. I'm not even sure what the benefit of using such a tiny bullet would be except maybe lighter recoil for people with shoulder injuries or the terminally sissy.
An ethical hunter will ensure that his bullet is of sufficient quality and size to kill game as cleanly and humanely as possible. Using the smallest possible projectile does not achieve that goal.