  While gravitational wave (GW) standard sirens from neutron star (NS) mergers
have been proposed to offer good measurements of the Hubble constant, we show
in this paper how a variation of the expanding photosphere method (EPM) or
spectral-fitting expanding atmosphere method, applied to the kilonovae (KNe)
associated with the mergers, can provide an independent distance measurement to
individual mergers that is potentially accurate to within a few percent. There
are four reasons why the KN-EPM overcomes the major uncertainties commonly
associated with this method in supernovae: 1) the early continuum is very
well-reproduced by a blackbody spectrum, 2) the dilution effect from electron
scattering opacity is likely negligible, 3) the explosion times are exactly
known due to the GW detection, and 4) the ejecta geometry is, at least in some
cases, highly spherical and can be constrained from line-shape analysis. We
provide an analysis of the early VLT/X-shooter spectra AT2017gfo showing how
the luminosity distance can be determined, and find a luminosity distance of
$D_L = 44.5\pm0.8$ Mpc in agreement with, but more precise than, previous
methods. We investigate the dominant systematic uncertainties, but our simple
framework, which assumes a blackbody photosphere, does not account for the full
time-dependent three-dimensional radiative transfer effects, so this distance
should be treated as preliminary. The luminosity distance corresponds to an
estimated Hubble constant of $H_0 = 67.0\pm 3.6$ km s$^{-1}$ Mpc$^{-1}$, where
the dominant uncertainty is due to the modelling of the host peculiar velocity.
We also estimate the expected constraints on $H_0$ from future KN-EPM-analysis
with the upcoming O4 and O5 runs of the LIGO collaboration GW-detectors, where
five to ten similar KNe would yield 1\% precision cosmological constraints.
