It depends.
There are many factors that come into play in making that decision.
The size of the deer. The maximum distance you will attempt a shot. The bullet type. The "presentation" of the shot, as in, standing broadside, quartering away, alert, grazing, etc.
I would definitely try it on small Texas whitetails, using a 60gr bullet, preferrably the Nosler.
I was looking at 6 does last Sunday evening, and none of them would weigh more than 90 pounds. I felt slightly overgunned with my Marlin 45-70. It would have worked fine, but I chose to not shoot, for several reasons. At that point, I was thinking that a smaller caliber, like my 6.5X55 would have been more reasonable.
If I was hunting larger mid-west whitetails or mulies, I probably would not consciencely choose the .223. (did I spell that word correctly?)