Semiconductor nanowires exhibit numerous capabilities to advance the development of future optoelectronic devices. Among the III-V material family, gallium phosphide (GaP) is an attractive platform with low optical absorption and high nonlinear susceptibility, making it especially promising for nanophotonic applications. However, investigation of single nanostructures and their waveguiding properties remains challenging owing to typically planar experimental arrangements. Here we study the linear and nonlinear waveguiding optical properties of a single GaP nanowire in a special experimental layout, where an optically trapped structure is aligned along its major axis. We demonstrate efficient second harmonic generation in individual nanowires and unravel phase matching conditions, linking between linear guiding properties of the structure and its nonlinear tensorial susceptibility. The capability to pick up single nanowires, sort them with the aid of optomechanical manipulation and accurately position pre-tested structures opens a new avenue for the generation of optoelectronic origami-type devices.