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We have undertaken a programme to observe emission lines of [S iv] 10.51, [Ne ii] 12.81, [Ne iii] 15.56, and [S iii] 18.71 μm in a number of extragalactic H ii regions with the Spitzer Space Telescope. Here we report our results for the nearly face-on spiral galaxy M83. A subsequent paper will present our data and analysis for another substantially face-on spiral galaxy M33. The nebulae selected cover a wide range of galactocentric radii ( RG). The observations were made with the infrared spectrograph in the short wavelength, high dispersion configuration. The above set of four lines is observed cospatially, thus permitting a reliable comparison of the fluxes. From the measured fluxes, we determine the ionic abundance ratios including Ne++/Ne+, S3+/S++ and S++/Ne+ and find that there is a correlation of increasingly higher ionization with larger RG. By sampling the dominant ionization states of Ne and S for H ii regions, we can approximate the Ne/S ratio by (Ne++ Ne++)/(S+++ S3+). Our findings of ratios that significantly exceed the benchmark Orion Nebula value, as well as a decrease in this ratio with increasing RG, are more likely due to other effects than a true gradient in Ne/S. Two effects that will tend to lower these high estimates and to flatten the gradient are first, the method does not account for the presence of S+ and second, S but not Ne is incorporated into grains. Both Ne and S are primary elements produced in α-chain reactions, following C and O burning in stars, making their yields depend very little on the stellar metallicity. Thus, it is expected that Ne/S remains relatively constant throughout a galaxy. We stress that this type of observation and method of analysis does have the potential for accurate measurements of Ne/S, particularly for H ii regions that have lower metallicity and higher ionization than those here, such as those in M33. Our observations may also be used to test the predicted ionizing spectral energy distribution (SED) of various stellar atmosphere models. We compare the ratio of fractional ionizations and versus with predictions made from our photoionization models using several of the state-of-the-art stellar atmosphere model grids. The overall best fit appears to be the nebular models using the supergiant stellar atmosphere models of Pauldrach, Hoffmann & Lennon and Sternberg, Hoffmann & Pauldrach. This result is not sensitive to the electron density and temperature range expected for these M83 nebulae. Considerable computational effort has gone into the comparison between data and models, although not all parameter studies have yet been performed on an ultimate level (e.g. in the present paper the stellar atmosphere model abundances have been fixed to solar values). A future paper, with the benefit of more observational data, will continue these studies to further discriminate how the ionic ratios depend on the SED and the other nebular parameters. [ABSTRACT FROM AUTHOR]