For floating ice, the mass of the ice is buoyantly supported by displacement of an equal mass of liquid water. As the SI units for mass were originally defined by the density of liquid water at STP, of 1 g/mL, when the ice melts, it supplies exactly as much volume as was previously displaced, so there is no change in level.
As it happens, seawater is denser than pure water, with a density averaging 1.088 g/mL, so if we were just talking about floating ice, 1 kg of floating ice would displace 1 kg of seawater, with a volume of 0.919 L, when the ice (mostly-pure water) melts, it supplies additional volume of 1 L, an increase of 8.8% over the submerged volume of the ice. This mixes with the saltwater to reduce salinity and lower the density, but the overall effect is a slight rise in water level.
Now forget about floating ice. Some ice is not floating. It rests upon surface rock. When it melts, 100% of its volume (eventually) flows down into the ocean. When land-ice melts, it has a far greater impact on sea level. But we only need to worry about land ice that rarely melts due to seasonal variation: ice found mainly in the glaciers and icepack on Greenland and Antarctica. Due to albedo and insulation effects, it is more difficult to melt land-ice than sea-ice, but it is indeed melting at rates unprecedented within the relatively short span of human observations.
(If a land-ice glacier calves an iceberg into the ocean, the immediate effect is to raise sea level by about 86% of the volume of the ice, with additional rise occurring as the ice melts.)