Do our continents lay on the mantle because solid silicon dioxide is less dense than its liquid form?
Even if the mantle was pure silica, it wouldn't melt at great depth because of the pressure and paucity of volatiles at that depth (which would act like fluxes, depressing the melting point). So it wouldn't behave like a solid phase floating on a chemically similar liquid phase.

For the most part the crust is not silica but a mix of silica and alumino-silicates of calcium, sodium and potassium, with traces of a whole lot of other stuff. The average specific gravity is about 2.8. On the other hand the mantle is predominantly a mixture of magnesium-iron silicates, with a little calcium, nickel, chromium, etc, and an average specific gravity of about 3.2.  So it is a case of silica-rich (not pure silica) rocks floating on much denser silicates, depleted in aluminium, and relatively depleted on silica. The biggest difference is the high iron content of the mantle.

I suspect confusion arises because many people are not clear about the distinction between quartz silica, which always has a specific gravity of 2.65, and silicates, where the density is highly variable between about 2.0 and 4.7.