At what depth below the Sun's surface does the density reach that of water? I'd like to write a short supplementary answer to A probe floating in Sun's surface but in order to do so I need to know at roughly what depth relative to the Sun's surface the mass density reaches 1 g/cm^3.

At the bottom of the photosphere the density seems to be either 1E-06 or 1E-09 g/cm^3 as discussed in the question What is the density profile within the Sun's photosphere? but how deep do you have to go below that to reach 1 g/cm^3?
The sun's density is 1 gm/cc at approximately 50% of the way down towards the core. if radius of Sun is R then at R/2 the density will be that of water. 

@astrosnapper's comment links to this answer in Physics SE. A reverse image search of that unsourced graphic leads to 


Thesis: Multi-spacecraft analysisof the solar coronal plasma
NASA: The Solar Interior
Whatever this is shown below
the link in that image http://www.sns.ias.edu/~jnb/SNdata/Export/BP2004/bp2004stdmodel.dat provides some raw data
the reference in that image is standard solar model BP2004 J. N. Bahcall and M. H. Pinsonneault, Phys. Rev. Lett. 92 (2004) 121301 What do we (not) know theoretically about solar neutrino fluxes? (also in ArXiv)