Can naphthenic acid be ionized in n-dodecane?
It would not ionize (though the correct term is that it won't dissociate, ie. the O-H bond in the acid won't cleave giving carboxylate anion).

It's because n-dodecane has no lone electron pairs, so it can't complex the hydrogen cation. Unless we're talking about extreme vacuum or plasma, there are no free $H^+$, it always exist in a form of a complex cation with a molecule of solvent or the like. But such solvent must have some electron donor: lone pair, double bond, etc. to form the complex.

The only other possible way here would be self-dissociation, where one of the substance molecules accept the hydrogen carion from the other one, but here it is also borderline impossible as the additionally protonated acid molecule would be very unstable and immidetely such auto-dissotiation would reverse.

The fact that the naphtalenic acid dissolves in n-dodecane is due to low polarity of both, that is acid molecules don't bind strongly to each other and n-dodecane molecules don't bind strongly to each other.