Current computing systems are facing two essential challenges: tremendous energy consumption due to the conventional Von Neumann architecture with low energy efficiency and environmental sustainability by depletion of nonrenewable materials, production of electronic waste, etc. One potential solution to simultaneously address these two issues is by brain-inspired neuromorphic computing with green electronic components, so that energy-efficient operation, sustainable material resources, and environmentally friendly disposals can be achieved. Such sustainable neuromorphic computing systems require hardware components not only capable of mimicking human neuron and synapse - the basic building block of biological neural networks but also made from natural organic materials such as polypeptides (proteins) and polysaccharides (carbohydrates) which are renewable, abundant in nature, biodegradable, eco-friendly, and with low-cost material and fabrication cost. In this paper, we report resistive switching random access memory (ReRAM) made from an encouraging natural organic carbohydrate material - fructose, for such emerging sustainable neuromorphic computing systems and neural networks. ReRAM has been proven to be a promising memory device technology for neuromorphic computing systems due to their ability to retain resistive states and respond to input signals in an analog or digital fashion, as well as fast speed, scaling and integration capability. Fructose resistive films incorporated with single wall carbon nanotubes (SWCNTs) were formed by a low-cost solution-based process and sandwiched between bottom and top electrode, a simple metal-insulator-metal structure which is analogous to a biological synapse with presynaptic neuron (top electrode), postsynaptic neuron (bottom electrode), and synaptic cleft (fructose-CNT film). The nonvolatile memory behaviors were demonstrated by modulating the conductance by voltage stimuli applied on the fructose-CNT ReRAM device, with excitatory current flow in the device being monitored. Characteristics including bipolar resistive switching, retention, endurance cycles, long-term potentiation and depression, dissolution in water, etc. are reported. All these results testify that fructose-CNT ReRAM devices are promising for energy-efficient and sustainable neuromorphic systems.