  We propose novel inflationary primordial gravitational wave (GW) spectral
shapes at interferometer-based current and future GW missions to test dark
matter (DM) production and baryogenesis via gravity-portal. We consider three
right-handed neutrinos (RHNs), the lightest among them is DM candidate while
the others participate in baryogenesis via leptogenesis. We find that future GW
detectors BBO, DECIGO, ET, for instance, are able to probe DM mass for $5\times
10^6\, {\rm GeV}<M_{\rm DM}<1.6\times 10^7$ GeV with a signal-to-noise ratio
(SNR) $>10$, along with the observed amount of baryon asymmetry due to
gravitational leptogenesis for heavy RHN mass $M_{\cal{N}}$ to be around
$8\times 10^{12}$ GeV. Employing Fisher matrix forecast analysis, we identify
the parameter space involving non-minimal coupling to gravity $\xi$, reheating
temperature of the Universe $T_{\rm rh}$ and DM mass $M_{\rm DM}$ where the GW
detector-sensitivities will be the maximum with the least error, along with SNR
$>10$. Finally, utilizing mock data for each GW detector, we perform MCMC
analysis to find out the combined constraints on the various microphysics
parameters. We also explore production of other cosmological relics such as QCD
axion relic as DM candidate, produced via gravity-portal in early universe. We
find that ET, for instance, can probe the decay constant of such DM candidates
($f_a$) as $10^9\,{\rm GeV}\lesssim f_a\lesssim 10^{14}\,{\rm GeV}$ for
misalignment angle $\theta_i\in[0.1,\pi/\sqrt{3}]$ and $\xi=1$ with SNR $>10$,
whereas this range decreases with the increase of non-minimal coupling. Thus
the upcoming GW missions will be able to test such non-thermal DM and
baryogenesis scenarios involving very high energy scales, which is otherwise
impossible to reach in particle physics experiments in laboratories.
