Is it possible to use the ice maker on a refrigerator without a water supply line?
Well I don't know what your budget is, but you could go ahead and get a keg and a compressed air tank and hook it up to the refrigerator! Easy 100psi, it'll make you 15 gallons of ice before refilling the keg (With water. The keg is to be filled with water. In case anyone missed that.)

Anyways now I have an awesome idea to use a fridge as a keg tap, if only I was in college where that'd be necessary. 

Parts would be:


1 gallon compressor, for example:


http://www.amazon.com/Senco-PC1010-1-Horsepower-1-Gallon-Compressor/dp/B0000AQK78/
http://www.amazon.com/DEWALT-D55140-1-Gallon-Trim-Compressor/dp/B000HZJMFM/

3 to 5 gallon keg, for example:


http://www.homebrewing.org/3-Gallon-Ball-Lock-Keg-W-Metal-Handles_p_3431.html
http://www.homebrewing.org/5-Gallon-Ball-Lock-Keg-W-Metal-Handles_p_3430.html

Barbed Liquid disconnect - http://www.homebrewing.org/Beer-Ball-Lock-Disconnect-Barbed_p_2915.html
Barbed Gas disconnect - http://www.homebrewing.org/Gas-Ball-Lock-White-Disconnect-Barbed_p_2916.html
Some food-safe 60psi tubing


Really easy version: Buy an entire homebrewing kit including tank and just put water in instead of beer, hook output up to fridge input.

Easy version: Set regulator of the compressor to 50psi, connect hose directly to barbs of gas in port of the keg. Hook the output port up to the fridge input. 

Hard version: Find a used compressor. Experiment with tightening the pressure switch until it turns off at 50psi. Remove compressor motor/pump assembly from tank, hook previous tank input up directly to keg, then keg output to fridge input.