As a revolutionary technology, reconfigurable intelligent surface (RIS) is favored by numerous industrialists and engineers, which can intelligently adjust the wireless communication environment. In our paper, the outage behaviors of RIS-assisted non-terrestrial networks with cognitive radio and non-orthogonal multiple access (NOMA) are analyzed. Particularly, we derive the exact expression for the outage probability of secondary users with perfect/imperfect successive interference cancellation (SIC) under the interference constraint of primary network. To obtain further insights, the asymptotic analysis for the outage probability and diversity order are derived at high signal-to-noise ratios. Moreover, numerical results are presented to verify the correctness of the theoretical derivation, and demonstrate the impacts of key parameters on outage behavior. The simulation results indicate that the combination of RIS and NOMA may significantly enhance performance of the considered system, especially compared to orthogonal multiple access scheme.