At what depth below the Sun's surface does the density reach that of water?
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)