Air Carrier Captain reported while taxiing on an icy ramp at night; they began to slide and shut down an engine shut down to stop aircraft.

Narrative

We landed Runway 32 in ITH; Tower was reporting Runways as condition 5;5;5; and there were no NOTAMS or reports of poor or NIL braking anywhere on the field. We were assigned taxi route C-A Ramp and I determined that braking action on Runways and Taxiways was good and relayed this to Tower after exiting C. Upon exiting A and taxing onto RAMP; it became apparent that we were sliding across a patch of ice and I was unable to slow the aircraft using brakes or to steer using the tiller or rudder inputs. At this point idle thrust was causing the aircraft to accelerate over the patch of ice and the nose was beginning to veer to the left. I shut down the left engine which stopped the aircraft from veering further left and prevented us from accelerating; but we continued to slide forward. I then shut down he right engine and with all thrust removed I was able to use the brakes to bring the aircraft to a full stop. We remained approximately 100 ft. on all sides of the aircraft from anything that we could have hit; no damage or injuries resulted from the slide. Ops dumped sand on the ramp and urged us to restart the engines and taxi to the Gate under our own power. I discussed with Dispatch and told them that I felt uncomfortable with that plan due to concerns of FOD ingestion and Ramp condition. Dispatch agreed and told Ops that we would need to find another way. Ops made the decision to deplane the passengers on foot; and the XX passengers were escorted across the ramp and into the Terminal (approx 50 yards). After more than an hour; on-call Maintenance arrived and towed the aircraft to the Gate.