Density comes into it if we are measuring buoyancy. The weight of the displaced water is equal to the weight of the buoyant object, and the ratio of volume submerged vs above water should give a way to determine density, but I don't have the precise method in my head right now.

However assuming all you want is the volume, you simply need to fully submerge the object (by applying enough downward force if its density is too low), and the displaced volume is equal to the objects volume. The displaced water will have to flow to the top of the container, so by measuring the rise in water level and multiplying with the base area of your container you can get it easily.

Just make sure you don't accidentally also measure your hand if you're pushing down the sphere.