When ice is floating it displaces it's own equivalent mass of water. The reason 9/10ths of the ice is submerged is that it's slightly less dense than water due to trapped air and that it expands when it freezes.
If floating ice melts it does not affect water levels. The Arctic is floating, it could melt completely and would have no effect on sea levels.
However, the Arctic accounts for only a tiny proportion of ice on the planet (0.01%), most is found in Antarctica (89.75%) and Greenland (9.97%); neither of which are floating. If the Arctic melts it disappears completely, if the Antarctic or Greenland ice melts it exposes land beneath (only the sea ice and ice sheets are over water).
When calculating changes to sea levels all these and more factors are taken into account.
The floating ice is made up from: The Ross Ice Shelf, Ronne-Filcher ice shelves, seasonal floating pack ice and the Arctic ice pack. Together they account for 2.1% of the ice on the planet. The remaining 97.9% is grounded or land based ice.
Arctic ice is melting at the rate of 600km³ per year and could be gone in 40 years, Greenland is melting at the rate of 220km³ per year at which rate it will last 13,000 years; Antractica is melting at the rate of 82km³ per year so would last for 300,000 years (figures based on average melting over the last 10 years and assume continued melting at the same rate into the future).
If the Arctic melts sea levels will not rise, if the Greenland ice melts completely sea levels will rise by 6.55 metres (21 feet), if Antracticea melts sea levels will rise by 73.41 metres (241 feet). If every bit of ice on the planet melts sea levels will rise by 80.32 metres (263 feet).
Current sea level rises are locally between -2 and +30mm per year with a global average of +3mm. The rate of rising is increasing, 100 years ago it was 1mm a year, in 50 years it's expected to be 6mm a year, by the end of the century it's expected that sea levels will have risen by 750mm and that the rate of rise then will be 12mm a year.