Inclusive transverse momentum spectra of η mesons in the range pT≈2-12 GeV/c have been measured at midrapidity (|η|<0.35) by the PHENIX experiment at RHIC in p+p,d+Au, and Au+Au collisions at √sNN=200 GeV. The η mesons are reconstructed through their η→γγ channel for the three colliding systems as well as through the η→π0π+π- decay mode in p+p and d+Au collisions. The nuclear modification factor in d+Au collisions, RdAu(pT)≈1.0-1.1, suggests at most only modest pT broadening ("Cronin enhancement"). In central Au+Au reactions, the η yields are significantly suppressed, with RAuAu(pT)≈0.2. The ratio of η to π0 yields is approximately constant as a function of pT for the three colliding systems in agreement with the high-pT world average of Rη/π0≈0.5 in hadron-hadron, hadron-nucleus, and nucleus-nucleus collisions for a wide range of center-of-mass energies (√sNN≈3-1800 GeV) as well as, for high scaled momentum xp, in e+e- annihilations at √s=91.2 GeV. These results are consistent with a scenario where high-pT η production in nuclear collisions at the Relativistic Heavy Ion Collider is largely unaffected by initial-state effects but where light-quark mesons (π0,η) are equally suppressed due to final-state interactions of the parent partons in the dense medium produced in Au+Au reactions.