Air carrier Captain reported temporary loss of control while taxiing due to ice and snow on the surface.

Narrative

During taxi in after a normal approach and landing; we were given taxi instructions by ramp control to enter the ramp at spot XX. While taxiing south on taxiway 1; I advised the FO to shut down the #2 engine as braking action was good. As we approached spot XX; ramp control instructed us to enter the spot and proceed straight into gate XX. Arriving at the spot; I made the right turn and lost sight of the taxi line due to snow and ice covering the line. The aircraft continued straight ahead on the taxi way and came to a stop due to the excessive ice/snow on the ground at idle thrust. We advised ramp control we were stopped and would need a few minutes to continue. I had the FO start the #2 engine and attempted to continue forward using minimum breakaway thrust. Unable to move the aircraft; we shut down the engines and requested a tug for tow to the gate. The tug arrived and we were towed to our gate after approximately 20 minutes without further incident. Cause: The ramp controller advised us that spot XX was open for taxi however; it did not appear to be plowed and was difficult to identify the depth of the ice as the whole ramp was covered. As a contributing factor; we were one of the first arrivals to the airport after the winter storm; so ramp operations were just getting back to normal. Even as the runway/taxiways had good breaking action; I should have kept the #2 engine running for taxi as the ramp area was still contaminated. The ramp controller apologized to us and stated he was told that spot XX taxi line was open. In the future; I will keep both engines running during contaminated ramp operations and ensure a good scan of the area before turning and recommend another entry point to the ramp if needed after significant winter weather.