  We propose a new method towards distinguishing the Dirac versus Majorana
nature of neutrino masses from the spectrum of gravitational waves (GWs)
associated with neutrino mass genesis. Motivated by the principle of generating
small neutrino masses without tiny Yukawa couplings, we assume generic seesaw
mechanisms for both Majorana and Dirac neutrino masses. For Majorana neutrinos,
we further assume a spontaneously broken gauged $U(1)_{B-L}$ symmetry,
independently of the type of Majorana seesaw mechanism, which gives a cosmic
string induced GW signal flat over a wide range of frequencies. For Dirac
neutrinos, we assume the spontaneous breaking of a $Z_2$ symmetry, the minimal
symmetry choice associated with all Dirac seesaw mechanisms, which is softly
broken, generating a peaked GW spectrum from the annihilation of the resulting
domain walls. In fact, the GW spectra for all types of Dirac seesaws with such
a broken $Z_2$ symmetry are identical, subject to a mild caveat. As an
illustrative example, we study the simplest respective type-I seesaw
mechanisms, and show that the striking difference in the shapes of the GW
spectra can help differentiate between these Dirac and Majorana seesaws,
complementing results of neutrinoless double beta decay experiments. We also
discuss detailed implications of the recent NANOGrav data for Majorana and
Dirac seesaw models.
