Inclusive transverse momentum spectra of eta mesons in the range p(T)approximate to 2-12 GeV/c have been measured at midrapidity (vertical bar eta vertical bar < 0.35) by the PHENIX experiment at RHIC in p+p,d+Au, and Au+Au collisions at root s(NN)=200 GeV. The eta mesons are reconstructed through their eta ->gamma gamma channel for the three colliding systems as well as through the eta ->pi(0)pi(+)pi(-) decay mode in p+p and d+Au collisions. The nuclear modification factor in d+Au collisions, R-dAu(p(T))approximate to 1.0-1.1, suggests at most only modest p(T) broadening ("Cronin enhancement"). In central Au+Au reactions, the eta yields are significantly suppressed, with R-AuAu(p(T))approximate to 0.2. The ratio of eta to pi(0) yields is approximately constant as a function of p(T) for the three colliding systems in agreement with the high-p(T) world average of R-eta/pi(0)approximate to 0.5 in hadron-hadron, hadron-nucleus, and nucleus-nucleus collisions for a wide range of center-of-mass energies (root sNN approximate to 3-1800 GeV) as well as, for high scaled momentum x(p), in e(+)e(-) annihilations at root s=91.2 GeV. These results are consistent with a scenario where high-p(T) eta production in nuclear collisions at the Relativistic Heavy Ion Collider is largely unaffected by initial-state effects but where light-quark mesons (pi(0),eta) are equally suppressed due to final-state interactions of the parent partons in the dense medium produced in Au+Au reactions.