Can a black hole with the density of water exist?
Applying a numerical density to a black hole isn't possible. The material inside the event horizon will fall to a "singularity" (or some other ultrahigh density state that we currently have no adequate theory to describe) on a relatively short timescale.

What you can do, is exactly what you have done, which is divide the gravitational mass of a black hole by the volume defined by a simple Euclidean estimate using the Schwarzschild radius$^{*}$.

If instead of substituting out the mass as you have done, you instead substitute out the Schwarzschild radius, then
$$ M = \left(\frac{3}{4\pi \rho}\right)^{1/2} \left(\frac{c^2}{2G}\right)^{3/2} = 1.4\times10^{8} \left(\frac{\rho}{\rho_w}\right)^{-1/2} \ M_{\odot},$$
where $\rho_w = 1000$ kg/m$^3$.

So, if you have a black hole of mass $1.4 \times 10^{8} M_{\odot}$, then it has the "density" of water if you calculate it like this.

There are lots of black holes with this mass or even higher that are situated in the centres of galaxies. The black hole at the centre of the Andromeda galaxy has a mass of about $10^{8} M_{\odot}$, so fits the bill perfectly.


Note that it is a vast simplification (and incorrect) to use $4\pi r_s^3/3$ as the volume inside the event horizon. There is in fact no uniquely defined volume; it depends on choice of coordinate system (see DiNunno &amp; Matzner 2008).