Of COURSE the surface area increases (as R^2), but volume increases (as R^3), so the RATIO of surface area to volume decreases as 1/R

Then it depends where the water comes from.

EITHER water is sweated out of a planetary interior from chondritic material differentiating and you get ~0.5% H2O by mass.

OR water arrives from a "late cometary veneer" and larger bodies attract more comets so water is again proportional to mass.

Or a bit of BOTH.

Either way don't seem to matter. Big worlds still have thicker oceans because the amount of water grows faster than the surface area to spread it over, so the ocean gets deeper.

Consider a Terrestrial world twice the radius of Earth (I know it would be too big to stop being a gas giant, but humour me, because it makes the maths easy). Call it Big-E.

Earth has a global 3-4 km ocean + ~ 50% more in the pores in the rocks, and as -OH compounds, so ~ 4.5-6 km global equivalent depth.

Big-E has 8 times the volume (and mass) [neglecting increased compressibility with a bigger planet, that just makes thing worse if you include it, anyway]

Big-E has 4 times the surface area.

So the equivalent depth of water on Big-E is 8 times that on Earth, divided by 4 times the area, so twice the depth. i.e. 9-12 km

Subtract the amount tied up in rocks, as per Earth -> 7.5-10 km ocean depth.

Imagine flooding Earth by a further 3.5 to 7 km of ocean. How much land would be left?

_____________

Now let's do "Earthlet" (which is a good proxy for Mars, except that the composition isn't quite right - more later on this).

Earthlet is half the diameter, 1/8 the mass & volume, 1/4 the surface area.

Equivalent depth of water here = half that of Earth = 2-3 km. Less amount taken up in rocks (1.5-2km) gives 0 to 1.5 km free water on Mars.

Caveats for Mars:

Planetary mean density is 3.9 cf 5.5 for earth, so Mars has more light elements. If that includes volatiles, then there may originally have been more.

D/H ratio suggests anywhere from 2-10 times enrichment, or 50% to 90% water loss to space. So modern day Mars has only 200m to 1.5 km available water equivalent depth, which is LESS THAN the amount that can be incorporated into the rock volume. (1.5 to 2 km)

Therefore we expect no free water at Mars' surface today, unless it originally had MUCH more volatiles by mass than earth and has lost as little as possible to space.

QED