  In seesaw mechanism, if right handed (RH) neutrino masses are generated
dynamically by a gauged $U(1)$ symmetry breaking, a stochastic gravitational
wave background (SGWB) sourced by a cosmic string network could be a potential
probe of leptogenesis. We show that the leptogenesis mechanism that facilitates
the dominant production of lepton asymmetry via the quantum effects of
right-handed neutrinos in gravitational background, can be probed by GW
detectors as well as next-generation neutrinoless double beta decay
($0\nu\beta\beta$) experiments in a complementary way. We infer that for a
successful leptogenesis, an exclusion limit on $f-\Omega_{\rm GW}h^2$ plane
would correspond to an exclusion on the $|m_{\beta\beta}|-m_1$ plane as well.
We consider a normal light neutrino mass ordering and discuss how recent
NANOGrav pulsar timing data (if interpreted as GW signal) e.g., at 95$\%$ CL,
would correlate with the potential discovery or null signal in $0\nu\beta\beta$
decay experiments.
