From my experience :6.5x55mm - excellent for deer; .30 cal. carbine - no way
.243 is commonly held as a good cartridge for whitetail,
probably wouldn't want anything smaller.
Part of what would be the minimum caliber would depend
upon how close you are to the animal, ex. .357 and .44
magnum pistol rounds are apparently ok if you're in close.
Used a high-power rifle for the first time this fall and shot
a very large doe. The deer went down so fast I didn't see
it fall. Used an old Swedish Mauser, 6.5x55mm. Hit it in
the backbone. Excellent cartridge.
The only other time I've used a rifle to kill a deer was an
M-1 carbine, .30 cal. I would not recommend that as a
deer cartridge. I was 15 at the time. Hit the deer in the
right front leg and left front leg, broke both bones. The
deer took off running on its two hind legs, looked like a
kangaroo leaping. I had to run it down about 150-200 yds
from where I shot it and then killed it with 2-3 more shots.
My father used the M-1 to deer hunt for a couple of years
and then he also quit. Said he was tired of tracking deer
1/2 mile after he shot them.