In this paper, we investigate the energy efficiency of a multiuser downlink Non-Orthogonal Multiple Acess (NOMA) system aided by an ambient backscatter device that modulates its own information by reflecting the incident signal coming from the NOMA transmitter. Because of the multiplicative operation at the ambient backscatter device when reflecting the transmitter's signal, the achievable sum rate of the system is not trivial. Hence, we first derive the information-theoretic achievable rate region for a discrete memoryless channel and, subsequently, for Gaussian channels. We then propose a joint optimization framework for maximizing the system energy efficiency as the tradeoff and ratio between the overall sum rate and the power consumption, under user minimum rate constraints. For this, we propose a modification that simplifies the non-convex optimization problem, which enables us to obtain the optimal reflection coefficient and power allocation policy analytically. Numerical results demonstrate the negligible impact of the introduced modification on the optimality of our solution. Remarkably, our results show that the ambient backscatter-aided NOMA significantly outperforms OMA: the relative gain increases with the number of receivers, reaching up to 14× improvement over OMA. At last, we show the pertinence of our solution also in the case of imperfect channel.