Am I correct in assuming that I can measure the voltage, use Ohm's Law, and get the resistance that easily?
Yes, assuming you know the current as well.
What is the smallest voltage value that the arduino will measure?
1.1 volts / 1024 AFAIK, ie about 1mV. So depending on the accuracy you need you would have to organise a constant current source that causes an appropriate voltage to appear across the ice.
use (40 mA, 50 mA,...)?
If you mean what current can the Arduino produce on a pin, we usually recommend 20mA as being the max working current. But that's not a CC source.
I guess you don't really need a CC source, but you do have to be able to measure the current, I think CC would be easier.
Rob