We examine neutron star properties based on a model of dense matter composed of B = 1 skyrmions immersed in a mesonic mean field background. The model realizes spontaneous chiral symmetry breaking nonlinearly and incorporates scale breaking of QCD through a dilaton vacuum expectation value that also affects the mean fields. Quartic self-interactions among the vector mesons are introduced on grounds of naturalness in the corresponding effective field theory. Within a plausible range of the quartic couplings, the model generates neutron star masses and radii that are consistent with a preponderance of observational constraints, including recent ones that point to the existence of relatively massive neutron stars M ~ 1.7 M⊙ and radii R ~ 12-14 km. If the existence of neutron stars with such dimensions is confirmed, matter at supranuclear density is stiffer than extrapolations of most microscopic models suggest.