6.5 Grendel for the win
Most capable of the low recoil rounds.
AR-15 has even less recoil and with an adjustable stock can be adjusted to fit her best. Most guns do not fit a recoil sensitive woman.

 

I would suggest a youth model bolt action .233. You have a lot more choices in types and grains of bullets in .223 than the 6mm arc or 22 arc. When shooting small calibers like this at deer it's even more important you are using the a bullet that provides the best terminal performance.

 

I’ll second Grem.

6.5 Grendel with either Hornady Black 123 ELD-M, or 123 SST’s.

 

Of the calibers mentioned I would go towards the 6mm. We have had a Remington short barrel rifle in the family for generations and it is our go to stalking gun and first gun for new shooters. (the short barrel versions are Loud but also light).
A round not mentioned is the 22-250. this is little to no recoil and extremely flat shooting and accurate. Definitedly worth looking into. Good luck and Kudos for reaching out to find a good rifle for your wife.

 

I was recently on a hunt in Illinois where rifle round choices are severely limited due to the government regulations there and had the opportunity to use a friends 350 legend versus the slug guns commonly used in that part of the country. I was really impressed with the 350. It had low recoil (seemed like less than my daughters 243) and I was able to shoot a buck at approximately 75 yards and he was DRT. May be a choice you haven't considered and would want to look into. I was so impressed, and rambled on about it so much my wife bought me one for Christmas. I'll still use my 308 down in South Texas as it has a longer effective range and the shots down there can be longer, but I will use the 350 at my East Texas lease.

 

6.5 Grendel or the 22-250. Cant go wrong with either one. I have been very impressed with the Grendel's performance.

 

In my opinion the .223 with the right bullet is plenty for deer, don't get so caught up in what cartridge is being and focus on quality bullets. A 77 grain tmk in .223 drops deer faster than most controlled expansion 30-06 loads.

I used to be skeptical of the .223 too but the tmk made a believer out of me.

 

I’d definitely recommend the 223. Plenty of power for deer and hogs out to normal hunting ranges. The Sierra 77 gr TMKs almost have a cult like following and are devastating on crittes. If you need some persuasion go read some of the over 10,000 posts on rockslide about its performance on game up to and including moose. As far as recoil goes the 223 is also the lowest of the bunch as well. There are lots of variables such as gun weight, bullet weight, etc, but generally speaking the Arc‘s and the Grendel are going to have about 20% more recoil.

 

6 ARC has done well on pigs for me so it will handle deer well. Any .22 is going to require precise shot placement with a well constructed bullet to be effective.

 

Check out the 7mmTCU..
Extremely easy to reload.
Good power.
Great for deer to 200

 

(Pigs too)