We explore the possibility of detecting Baryon Acoustic Oscillations (BAO) solely from gravitational wave observations of binary neutron star mergers with third generation (3G) gravitational wave (GW) detectors like Cosmic Explorer and the Einstein Telescope. These measurements would provide a new independent probe of cosmology. The detection of the BAO peak with current generation GW detectors (solely from GW observations) is not possible because i) unlike galaxies, the GW mergers are poorly localized and ii) there are not enough merger events to probe the BAO length scale. With the 3G GW detector network, it is possible to observe $\sim \mathcal{O}(1000)$ binary neutron star mergers per year localized well within one square degree in the ...