Can a black hole with the density of water exist? During an outreach event, the speaker mentioned the fact that black holes weren't necessarily dense objects and that black holes of any density could exist, even black holes with the density of water. 

Now indeed, the schwarzschild radius of such a black hole would be
$R = c\sqrt{\frac{3}{8 \pi G \rho}}$
with $\rho$ the density of water. 

But could such a non-dense black hole exist in practice? If not, what would prevent it from forming or from being stable?
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).