Memristive stateful logic for Boolean computers and memristive neural networks for neuromorphic computers are two distinct emerging applications enabled by memristors in future computing. Interestingly, they both utilize an identical crossbar array platform, suggesting their simultaneous implementation is possible. Herein, a new methodology combining the two technologies to create synergy in neuromorphic computing is proposed. A genetic algorithm in the memristive neural network is introduced, where the stochastic stateful logic realizes the required mutation and crossover operators. Under optimized genetic evolution conditions with the fittest selection algorithm, without any backpropagation circuits, the modified national institute of standards and technology dataset recognition accuracy of 90.1% for a 784 x 100 size network is anticipated, which is comparable to 93.9% accuracy using a conventional deep neural network.