Climate change & sea level rise - confused how melting ice can cause water levels to rise?

There are many people out there, both people who believe in climate change and people who might not, who don't understand at least one important aspect about it. When talking to a good friend of mine recently, while he was agreeing that what we are doing to out planet can cause problems, he told me that the rising sea levels because of melting ice is still sort of bull. Reason? Because ice is less dense than water, and as such, when it melts, the sea levels shouldn't rise. When water expands when it freezes, one ounce of frozen water has a larger volume than one ounce of liquid water. So the water level remains the same if the ice melts.

- Any floating object displaces a volume of water equal in weight to the object's mass. - Any submerged object displaces a volume of water equal to the object's volume. - Freshwater is not as dense as saltwater.

So, taking these things into account, what about the sea level rise if the ice starts melting? There's around 1.3 billion cubic kilometers of ocean water, and 660,000 cubic kilometers of floating sea ice. If both the ice and sea water were both either fresh of salt water, the melting ice wouldn't do a thing. When the ice melts, the sea levels would remain exactly the same. However, as the density of sea water and ice is different, the sea levels rise around 2.6% when the floating ice melts. That by itself is nothing, it would account for only around 4 centimeter sea level rise. However, if we also add the non-floating ice, which is around 50 times higher, the sea level rise become significant.

I'm no scientist, but this makes sense, doesn't it?