  The Thomas-Fermi approach to galaxy structure determines selfconsistently the
fermionic warm dark matter (WDM) gravitational potential given the distribution
function f(E). This framework is appropriate for macroscopic quantum systems:
neutron stars, white dwarfs and WDM galaxies. Compact dwarf galaxies follow
from the quantum degenerate regime, while dilute and large galaxies from the
classical Boltzmann regime. We find analytic scaling relations for the main
galaxy magnitudes as halo radius r_h, mass M_h and phase space density. The
observational data for a large variety of galaxies are all well reproduced by
these theoretical scaling relations. For the compact dwarfs, our results show
small deviations from the scaling due to quantum macroscopic effects. We
contrast the theoretical curves for the circular velocities and density
profiles with the observational ones. All these results are independent of any
WDM particle physics model, they only follow from the gravity interaction of
the WDM particles and their fermionic nature. The theory rotation and density
curves reproduce very well for r < r_h the observations of 10 different and
independent sets of data for galaxy masses from 5x10^9 Msun till 5x10^{11}
Msun. Our normalized circular velocity curves turn to be universal functions of
r/r_h for all galaxies and reproduce very well the observational curves for r <
r_h. Conclusion: the Thomas-Fermi approach correctly describes the galaxy
structures (Abridged).
