Low Earth Orbit (LEO) satellite networks are expected to play a crucial role in providing high-speed Internet access and low-latency communication worldwide. However, some challenges can affect the performance of LEO satellite networks. For example, they can face energy and spectral efficiency challenges, such as high power consumption and spectral congestion, due to the increasing number of satellites. Furthermore, mobile ground users tend to operate with low directive antennas, which pose significant challenges in closing the LEO-to-ground communication link, especially when operating at a high-frequency range. To overcome these challenges, energy-efficient technologies like reconfigurable intelligent surfaces (RIS) and advanced spectrum management techniques like non-orthogonal multiple access (NOMA) can be employed. RIS can improve signal quality and reduce power consumption, while NOMA can enhance spectral efficiency by sharing the same resources among multiple users. This paper proposes an energy-efficient RIS-assisted downlink NOMA communication for LEO satellite networks while ensuring the quality of services. The proposed framework simultaneously optimizes the NOMA transmit power of the LEO satellite and the passive beamforming of RIS, considering the assumption of imperfect successive interference cancellation. Due to the nature of the considered system and optimization variables, the energy efficiency maximization problem is non-convex. In practice, obtaining the optimal solution for such problems is very challenging. Therefore, we adopt alternating optimization methods to handle the joint optimization in two steps. In step 1, for any given phase shift vector, we calculate satellite transmit power towards each ground terminal using the Lagrangian dual method. Then, in step 2, given the transmit power, we design passive beamforming for RIS by solving the semi-definite programming. We also compare our solution with a benchmark framework having a fixed phase shift design and a conventional NOMA framework without involving RIS. Numerical results show that the proposed optimization framework achieves 21.47% and 54.9% higher energy efficiency compared to the benchmark and conventional frameworks.