This is an experiment you can do yourself in about 30 seconds.
Use a glass so you can see to the bottom. Fill it with ice to the top. I don’t care how big the pieces are. Now, start adding water until the water reaches the top.
How much ice is resting on the bottom?
Here's the answer: When the water reaches about 92% of the height of the block.
If you were filling the bathtub with seawater the block would float when the water reached 89% of the height of the block.
The reason why some glaciers in Antarctica rest on the bottom of the ocean is that the depth of the ocean in that location is less than 89% of the average height of the glacier.
By the way, if you put a 10-lb block of ice in your bathtub and fill the tub to less than the height at which the block floats, then when the block melts the water level will be higher.