I have the same tiny little white particles in my refrigerator water.
They appear to be mineral deposits. I even considered taking out the
water canister to flush it, but concluded that because of the way it's
built (only small inlet/outlet holes), that wouldn't work either.
Short of buying a new canister and installing an inline filter, I
couldn't come up with a solution.

I also recently flushed my water heater and all the sediment was white
flakes too, though much larger and higher quantity.