Could Global Climate Change increase the average moisture in the air?
Out of the total water on Earth, only a tiny fraction is held in the atmosphere at any given time.  Therefore, the direct effect of melting land or sea ice will not result in an increase in specific humidity.

However, warm air can contain more water than cold air, according to the Clausius-Clapeyron equation.  Therefore, the climate warming that causes the ice caps to melt, will also result in an increase in average specific humidity.  This is (approximately) an exponential relation (I don't have access to my textbooks right now, but googling saturation vapour pressure water equation yields many forms; I do not include them here because none I found use SI units).


Saturation vapour pressure of water.  Source: Chemguide.

See also the (not very good) Wikipedia article on Vapour pressure of water.