GE side by side CAN17 ice maker...white ''finger assembly'' that pivots has broken two fingers, and the ice cubes are stuck in the cube dispenser. Any ideas why this happens, and how to fix it without replacing the whole ice maker assembly?

Several possibilities, but check first for cubes that are too small (means too little water is coming in during ice maker fill) and those cubes fracture into smaller bits when they are dispensed.

Another possibility is the cube/crushed plate inside the dispenser is stuck. This is a curved plastic piece that lies next to the crusher blades inside the dispenser head of the ice storage bin. For cubes this place must be retracted (swing away from) the crushed fingers to allow the cubes to pass without being broken up.

Check the the part called the cube rake. Its usually a white plastic piece with fingers that point to each cube mold. Its suppose to push the cubes out during the cube dump cycle. It sounds like a few of the fingers may be broken and not turning properly on the shaft to extract the cubes out of the icemaker. The rake should push the cubes out during the cycle.

Ice maker bad or the ice maker water valve mounted on the same bracket as the dispenser is bad. These frigs are prone to bad ice makers after a few years.

take the bin out and check for a jammed ice cube in the ice cube making mechanism. Many times they have a rotating finger that gets stuck on a cube that does not drop entirely. Hope this helps

two common problems are the spring comes off the feeler arm or the ice cubes jam on the sweeper the plastic arm with fingers on the edge of the ice maker

Remove the ice bin. Look at the back wall, behind where the bin was. You should see a rectangular hole, about 1.5 x 1 inches near the bottom of where the ice bin was. There should be a white plastic part inside the hole. This plastic part should be able to be lifted up with your finger, and drop down again when you release it. If is does not move, it may be frozen, or the electrical solenoid connected to it may have melted. If the part moves freely, just replace the ice bucket and try to dispense cubes again. (the metal arm sticking out the back of the ice bucket might have gotten on the wrong side of the white plastic part in back)

FINGERS THAT EJECT THE CUBES WAS JAMMED ON WHITE PLASTIC. REMOVED ICE MAKER THAWED CUBES IN THE TREY AND PRYED AWAY THE WHITE PLASTIC COVER. RE-INSTALLED 

There is a solinoid in the freezer section by the motor that turns the ice auger. Test it by selecting cubes and with the ice bucket out, open the door, hold the door switch so the lights turn out and push the dispenser button. It should make a loud snap and travel upward. If not then I would say you have a bad solinoid. If it does, then I would look at the bucket. the metal rod that runs along the side of the bucket opens a flap in the bucket that is supposed to let the full cubes drop. look for anything that might cause it to be hanging up.

You may have a broken solenoid inside the motor assembly.  The solenoid controls the arm for cubed or crushed ice.  The only way to tell is remove the ice dispenser motor assembly and look at the solenoid.  If the solenoid is broken, replace it and everything should be ok.