  We analyzed the thermal emission from the entire surface of the millisecond
pulsar PSR J0437-4715 observed in the ultraviolet and soft X-ray bands. For
this, we calculated non-magnetized, partially ionized atmosphere models of
hydrogen, helium, and iron compositions and included plasma frequency effects
that may affect the emergent spectrum. This is particularly true for the
coldest atmospheres composed of iron (up to a few percent changes in the soft
X-ray flux). Employing a Markov chain Monte Carlo method, we found that the
spectral fits favour a hydrogen atmosphere, disfavour a helium composition and
rule out iron atmosphere and blackbody models. By using a Gaussian prior on the
dust extinction, based on the latest 3D map of Galactic dust, and accounting
for the presence of hot polar caps found in previous work, we found that the
hydrogen atmosphere model results in a well-constrained neutron star radius
$R_\mathrm{NS} = 13.6^{+0.9}_{-0.8}~\mathrm{km}$ and bulk surface temperature
$T^{\infty}_\mathrm{eff}=\left(2.3\pm0.1\right)\times10^5~\mathrm{K}$. This
relatively large radius favours a stiff equation of state and disfavours a
strange quark composition inside neutron stars.
