Its a big number, difficult to estimate I assume. But is there a peer reviewed estimation (at current conditions) of the volume of daily or annual increase of water?
I don't know if this helps, but since you have no response yet perhaps I can offer some thoughts.Your question is problematic due to the fact that volume itself is variable according to the temperature of the water. If you look at the Nasa site concerning sea level change:
Sea level rise is caused primarily by two factors related to global warming: the added water from melting ice sheets and glaciers and the expansion of sea water as it warms.
(some nice graphics here showing change amounts)
This expansion due to warming literally changes the volume of the water, adding to the sea level rise. So glacial melt may be measured or estimated, but the change in sea level caused by this volume of water is variable; over time it will change temperature and expand. But this isn't straightforward either. Nasa again:
Thermal expansion of seawater can be the product of regional phenomena, such as El Niño, the periodic warming of the eastern tropical Pacific. But some of these regional cycles so far show no direct link to long-term global climate change—despite, at times, independently exerting a powerful short-term influence on global climate.
So bottom line is that it is easier to measure the amount of change, then to calculate direct figures concerning how much actual water was involved in causing that change. Hope this helped.
Update: There is some specific data on another page of the Nasa site concerning the volume of water being generated by the melt of the Greenland Ice Sheet:
The estimated total loss is in the range of more than 200 to more than 300 gigatons per year (1 gigaton is approximately 264 billion gallons of water. Melting 365 gigatons of ice would add 1 millimeter to global sea level; there are 25.4 millimeters in an inch).
This does give you some approximations in gallons per year of one of the main contributors to the sea level rise.