Using the OSIRIS tunable narrowband imager on the 10.4 m GTC (La Palma), we have mapped the SAB(rs)cd galaxy NGC 6946 over a ~7.3 × 7.5 arcmin2 field in the emission lines of the [S II] λλ6717, 6731 doublet, and in Hα. From these maps we have produced catalogs of the Hα luminosities and effective radii of 557 H II regions across the disk, and derived the [S II] emission line ratios of 370 of these. The Hα observations were used to derive the mean luminosity-weighted electron densities for the regions of the sample, while the [S II] line ratios allowed us to derive values of the in situ electron densities in the denser zones from which the major fraction of the radiation in these lines is emitted for 58 of the regions. This is by far the largest data set of its kind for a single galaxy. A classical two-phase model is used to derive the filling factors of the regions. We find that although the mean electron density decreases with the square root of the radius of the regions, the in situ density is essentially independent of this radius. Thus the filling factor falls systematically, as the radius and the luminosity of the regions increases, with a power law of exponent -2.23 between filling factor and radius. These measurements should enhance the perspectives for more refined physical models of H II regions.