GaN has been utilized in optoelectronics for two decades. However, p-type doping still remains crucial for realization of high performance GaN optoelectronics. Though Mg has been used as a p-dopant, its efficiency is low due to the formation of Mg-H complexes and/or structural defects in the course of doping. As a potential alternative p-type dopant, Cu has been recognized as an acceptor impurity for GaN. Herein, we report the fabrication of Cu-doped GaN nanowires (Cu:GaN NWs) and their p-type characteristics. The NWs were grown vertically via a vapor-liquid-solid (VLS) mechanism using a Au/Ni catalyst. Electrical characterization using a nanowire-field effect transistor (NW-FET) showed that the NWs exhibited n-type characteristics. However, with further annealing, the NWs showed p-type characteristics. A homo-junction structure (consisting of annealed Cu:GaN NW/n-type GaN thin film) exhibited p-n junction characteristics. A hybrid organic light emitting diode (OLED) employing the annealed Cu:GaN NWs as a hole injection layer (HIL) also demonstrated current injected luminescence. These results suggest that Cu can be used as a p-type dopant for GaN NWs.