In this paper, a combination of energy harvesting (EH) and cooperative nonorthogonal multiple access (NOMA) has been proposed for full-duplex (FD) relaying vehicle-to-vehicle (V2V) networks with two destination nodes over a Rayleigh fading channel. Different from previous studies, here both source and relay nodes are supplied with the energy from a power beacon (PB) via RF signals, and then use the harvested energy for transmitting the information. For the extensive performance analysis, the closed-form expressions for the performance indicators, including outage probability (OP) and ergodic capacity of both users, have been derived rigorously. Additionally, the effect of various parameters, such as EH time duration, residual self-interference (RSI) level, and power allocation coefficients, on the system performance has also been investigated. Furthermore, all mathematical analytical results are confirmed by Monte-Carlo simulations, which also demonstrate the optimal value of EH time duration to minimize the OP and maximize the ergodic capacity of the proposed system.