  Steadily accreting white dwarfs (WDs) are efficient sources of ionization and
thus, are able to create extended ionized nebulae in their vicinity. These
nebulae represent ideal tools for the detection of accreting WDs, given that in
most cases the source itself is faint. In this work, we combine radiation
transfer simulations with known H and He accreting WD models, providing for the
first time the ionization state and the emission line spectra of the formed
nebulae as a function of the WD mass, the accretion rate and the chemical
composition of the accreted material. We find that the nebular optical line
fluxes and radial extent vary strongly with the WD's accretion properties,
peaking in systems with WD masses of 0.8 - 1.2 $\rm~M_{\odot}$. Projecting our
results on the 'BPT' diagnostic diagrams, we show that accreting WDs nebulae
possess characteristics distinct from those of H II-like regions, while they
share similar line ratios with the galactic low-ionization emission-line
regions. Finally, we compare our results to the relevant constraints imposed by
the lack of ionized nebulae in the vicinity of supersoft X-ray sources (SSSs)
and Type Ia supernova remnants - sources which are related to steadily
accreting WDs. The large discrepancies uncovered by our comparison rule out any
steadily accreting WD as a potential progenitor of the studied remnants and
additionally require the ambient medium around the SSSs to be less dense than
0.2 $\rm~cm^{-3}$. We discuss possible alternatives that could bridge the
incompatibility between the theoretical expectations and the relevant
observations.
