Nope, the rise in water shows exclusively the volume. Density is derived using volume and mass, with mass being measurable by weighing. The only factor density plays directly is indeed floating, but you can just push the sphere under with a thin rod without significantly impacting the measurement.

It's generally simple - water has some volume, sphere has some volume. Assuming a cylindrical container, the base of the cylinder is known, and the volume of it is equal to the volume of water, which means you can derive the height (or vice versa). If you now add the sphere, the sphere pushes out the water, which means you add the volume of the sphere to the water - and if you measure the height, you can calculate the total volume of water+sphere, and subtract the initial volume of water to isolate the volume of the sphere.