Having the right shooting skills, good judgement and enough self control, a 223/222 should be fine for deer(including stags).
However, only once these things above are sorted, try to go for something bigger. a 243 will offer both Varmint and Big game abilities.
I've shot 8-10 deer (including stags) with a 223 Rem and now my 223 Max. Personally, I believe that having enough self control to shoot or not shoot, consider what shot options are presented at the time(including bullet path/angle once in side the animal) and if you can be a very sneaky bastard while stalking, get into in. Otherwise, go bigger!
This was a summer time stag, stalked into about 10-12 meters and shot though the front of chest, down though the heart, with a 60gr Nosler Partition, going very, very, very fast(3550fps) out of my 223Max. It didn't take another step.
Bullet construction is pretty important too. The Partition is a great, well constructed bullet for deer, but can struggle to group in some rifles. I over came this by making it go faster.
Also had a few around 50-70 meters on spikers. Again, waiting for the correct shot to present itself and understanding that not shooting the deer in front of you is actually OK. I've walked away from animals after deciding that I cannot make the right shot at the time.
For the record, I do most (90%) of my hunting with 270WSM (140 or 150gr bullet) to ensure that I have a greater range of options when I find and animal, this includes; greater distance, better in the wind and if shooting just before dark, preventing the animal from running as far after a shot and not been able to recover it in the dark.
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