Initially the surface area of ice melting in air and ice melting in water is the same, but as ice melts in air a thin layer of water results, which absorbs some of the heat from the air and slightly insulates the remaining ice.
When you melt an ice cube in a cup of water it is exposed to both air and water. The part of the ice cube in the water melts faster than the ice in the air, but as the ice cube melts, it sinks further down. If you support the ice to prevent the ice cube from sinking, you could see the part of the ice in the water would melt more quickly than the part in the air.
Other factors come into play. If the air is blowing across the ice cube, then the increased circulation may allow the ice to melt faster in air than in water. If the air and water are different temperatures, the ice may melt more quickly in the medium with the higher temperature.