Does ice increase the humidity of its surroundings?
It depends on the current humidity. 

Water vapor certainly does evaporate from ice, as you can see in a vapor pressure chart. If the humidity is 100%, and the air temperature is the same as that of the ice, then the evaporation and condensation are in equilibrium; i.e. water evaporates as fast as it condenses, so there is no net change in humidity.

However, if the ice is warmer than the air, or the air has less than 100% moisture content at that temperature, then ice will evaporate more than it condenses, raising the humidity.

Conversely, if the ice is colder than the atmosphere, it can lower the humidity.