We present mid-infrared Infrared Space Observatory Short-Wavelength Spectrometer (ISO-SWS) observations of the fine-structure emissions lines [Ne II] 12.8 μm, [Ne III] 15.6 μm, [Ne III] 36.0 μm, [Ar II] 6.99 μm, [Ar III] 8.99 μm, [S III] 18.7 μm, [S III] 33.5 μm, and [S IV] 10.5 μm and the recombination lines Brα and Brβ in a sample of 112 Galactic H II regions and 37 nearby extra-Galactic H II regions in the LMC, SMC, and M33. We selected our sources from archival ISO-SWS data as those showing prominent [Ne II] 12.8 μm or [Ne III] 15.6 μm emissions. The Galactic sources have a wide range in galactocentric distance (0 kpc R_gal 18 kpc), which enables us to study excitation and metallicity variations over large Galactic scales. We detect a steep rise in the [Ne III] 15.6 μm/[Ne II] 12.8 μm, [Ar III] 8.99 μm/[Ar II] 6.99 μm, and [S IV] 10.5 μm/[S III] 33.5 μm excitation ratios from the inner Galaxy outward, and a moderate decrease in metallicity, from ~2 Z in the inner Galaxy to ~1 Z in the outer disk. The extra-Galactic sources in our sample show low gas density, low metallicity, and high excitation. We find a good correlation between [Ne III] 15.6 μm/[Ne II] 12.8 μm and [Ar III] 8.99 μm/[Ar II] 6.99 μm excitation ratios in our sample. The observed correlation is well reproduced by theoretical nebular models that incorporate new-generation wind-driven non-LTE model stellar atmospheres for the photoionizing stars. In particular, the non-LTE atmospheres can account for the production of [Ne III] emission in the H II regions. We have computed self-consistent nebular and stellar atmosphere models for a range of metallicities (0.5-2 Z). We conclude that the increase in nebular excitation with galactocentric radius is due to an increase in stellar effective temperature (as opposed to a hardening of the stellar spectral energy distributions due to the metallicity gradient). We estimate an integrated [Ne III] 15.6 μm/[Ne II] 12.8 μm ratio for the Galaxy of 0.8, which puts it well inside the range of values for starburst galaxies. The good fit between observations and our models support the conclusion of Thornley and coworkers that the low [Ne III] 15.6 μm/[Ne II] 12.8 μm ratios observed in extra-Galactic sources are due to global aging effects.