Nearby early-type galaxies with ionized gas.IV. Origin and powering mechanism of the ionized gas

[ABRIDGED] With the aim of constraining the source of excitation and the origin of the ionized gas in early-type galaxies (ETGs), we analyzed optical spectra of a sample of 65 ETGs mostly located in low density environments. Optical emission lines are detected in 89% of the sample. The incidence and strength of emission do not correlate either with the E/S0 classification, or with the fast/slow rotator classification. Comparing the nuclear r<r_e/16 line emission with the classical [OIII]/Hb vs [NII]/Ha diagnostic diagram, the galaxy activity is so classified: 72% are LINERs, 9% are Seyferts, 12% are Composite/Transition objects, and 7% are non-classified. Seyferts have young luminosity-weighted ages (<5 Gyr), and are significantly younger than LINERs and Composites. Seyferts excluded, the spread in the ([OIII], Ha or [NII]) emission strength increases with the galaxy central velocity dispersion. The [NII]/Ha ratio decreases with increasing galacto-centric distance, indicating either a decrease of the nebular metallicity, or a progressive "softening" of the ionizing spectrum. The average oxygen abundance of the ionized gas is slightly less than solar, and a comparison with the results obtained in Paper III from Lick indices reveals that it is ~0.2 dex lower than that of stars. Conclusions: the nuclear emission can be explained with photoionization by PAGB stars alone only in ~22% of the LINERs/Composite sample. On the other hand, we can not exclude an important role of PAGB star photoionization at larger radii. For the major fraction of the sample, the nuclear emission is consistent with excitation from a low-accretion rate AGN, fast shocks (200 -500 km/s) in a relatively gas-poor environment (n< 100 cm^-3), or coexistence of the two. The derived nebular metallicities suggest either an external origin of the gas, or an overestimate of the oxygen yields by SN models.