  In type I seesaw models, the right-handed neutrinos are typically
super-heavy, consistent with the generation of baryon asymmetry via standard
leptogenesis. Primordial gravitational waves of cosmological origin provides a
new window to probe such high scale physics, which would otherwise be
inaccessible. By considering a {\em global} $U(1)_{B-L}$ extension of the type
I seesaw model, we explore the connection between the heaviest right-handed
neutrino mass and primordial gravitational waves arising from the dynamics of
global cosmic string network. As a concrete example, we study a global
$U(1)_{B-L}$ extension of the Littlest Seesaw model, and show that the
inevitable GW signals, if detectable, probe the parameter space that can
accommodate neutrino oscillation data and successful leptogenesis, while
respecting theoretical constraints like perturbativity of the theory. Including
CMB constraints from polarization and dark radiation leaves a large region of
parameter space of the model, including the best fit regions, which can be
probed by GW detectors like LISA and ET in the near future. In general, the GW
detectors can test high scale type I seesaw models with the heaviest
right-handed neutrino mass above $2.5 \times 10^{14}$ GeV, assuming the
perturbativity, and $7 \times 10^{13}$ GeV assuming that the coupling between
the heaviest right-handed neutrino and the $U(1)_{B-L}$ breaking scalar is less
than unity.
