No, it's proportional to the mass.

http://en.wikipedia.org/wiki/Schwarzschild_radius

The Schwarzschild radius of a mass is not the radius of the object. It is the radius below which if a non-rotating spherically symmetric distribution with that mass were contained within it it would constitute a black hole. Frank's answer is correct.

Now let's get back to our originally scheduled broadcast and purpose for this forum: Wolfram technologies. ;-)

Ok, I think that I can give you a more sensible answer.

Here is the expression for the Schawrtzschild radius in terms of the Mass:

$r=\frac{2 G M}{c^2}$

Now substitute for the mass in terms of the density ( $M=\frac{4 \pi r^3}{3}$) and get:

$r=\frac{8 \pi G \rho r^3}{3 c^2}$

Now solve for r,

Solve[r == (8 G \[Pi] r^3 \[Rho])/(3 c^2), r]

and get

{{r -> 0}, 
{r -> -((c Sqrt[3/(2 \[Pi])])/( 2 Sqrt[G] Sqrt[\[Rho]]))},
 {r -> (c Sqrt[3/(2 \[Pi])])/( 2 Sqrt[G] Sqrt[\[Rho]])}}

and the last solution is clearly the physical one you are looking for:

$\frac{\sqrt{\frac{3}{2 \pi }} c}{2 \sqrt{G} \sqrt{\rho }}$

Is this the correct interpretation of your question? I hope this helps...

See my revised answer above.... By the way, if you forget the expression for the Schwartzschild radius you can get it from

FormulaData[{"BlackHoleEventHorizonRadius", "Standard"}]

Apropos of my computation above here is the value one gets if one uses the density of water:

UnitConvert[
  (Quantity["SpeedOfLight"] Sqrt[3/(2 \[Pi])])/(2 Sqrt[
      Quantity["GravitationalConstant"]] Sqrt[
      Quantity[1, "Grams"/"Centimeters"^3]]), "Kilometers"]/
 UnitConvert[PlanetData["Earth", "Radius"], "Kilometers"]

which gives

6.297 10^4

times the radius of the earth...

It's worth noting in passing that my simple calculation above is rather unrealistic. That the density is uniform throughout the sphere (assuming that the radius is less than the Schawrtzschild radius so the object is not a black hole and so that we can then discuss its internals) is not physically reasonable. Assuming that the sphere is made of the same "stuff" throughout, one needs an equation of state to determine what its steady state composition is--particularly what the density is as a function of radius, and this will not be constant.

Frank that thank you was for you as well.