A central endeavour in bioelectronics is the development of logic elements to transduce and process ionic to electronic signals. Motivated by this challenge, we report fully monolithic, nanoscale logic elements featuring n- and p-type nanowires as electronic channels that are proton-gated by electron-beam patterned Nafion. We demonstrate inverter circuits with state-of-the-art ion-to-electron transduction performance giving DC gain exceeding 5 and frequency response up to 2 kHz. A key innovation facilitating the logic integration is a new electron-beam process for patterning Nafion with linewidths down to 125 nm. This process delivers feature sizes compatible with low voltage, fast switching elements. This expands the scope for Nafion as a versatile patternable high-proton-conductivity element for bioelectronics and other applications requiring nanoengineered protonic membranes and electrodes.