How can I prevent this ice build up on my driveway/walkway?
I'm not a professional, and I have very little experience. But:

Could you adjust the slope of your gutter?

Currently, rain rolls north off the roof, gets to the gutter, half goes east and half goes west. If you lower the north-east corner of the gutter (and gently slope the whole gutter toward it) a majority of the rain would head down the east spout instead of onto your driveway.