The radius R1.4 of neutron stars (NSs) with a mass of 1.4 times solar mass has been extracted consistently in many recent studies in the literature. Using representative R1.4 data, we infer high-density nuclear symmetry energy Esym(ρ) and the associated nucleon specific energy E0(ρ) in symmetric nuclear matter (SNM) within a Bayesian statistical approach using an explicitly isospin-dependent parametric equation of state (EOS) for nucleonic matter. We found the following. (1) The available astrophysical data can already significantly improve our current knowledge about the EOS in the density range of ρ0-2.5ρ0. In particular, the symmetry energy at twice the saturation density ρ0 of nuclear matter is determined to be Esym(2ρ0)= 39.2 (+12.1 -8.2)MeV at a 68% confidence level. (2) A precise measurement of R1.4 alone with a 4% 1σ statistical error but no systematic error will not greatly improve the constraints on the EOS of dense neutron-rich nucleonic matter compared to what we extracted from using the available radius data. (3) The R1.4 radius data and other general conditions, such as the observed NS maximum mass and causality condition, introduce strong correlations for the high-order EOS parameters. Consequently, the high-density behavior of Esym(ρ) inferred depends strongly on how the high-density SNM EOS E0(ρ) is parameterized, and vice versa. (4) The value of the observed maximum NS mass and whether it is used as a sharp cutoff for the minimum maximum mass or through a Gaussian distribution significantly affects the lower boundaries of both E0(ρ) and Esym(ρ) only at densities higher than about 2.5ρ0.