We have investigated the magnetic properties, the electronic structure, and the formation energy of wurtzite GaN nanowires (NWs) with gallium vacancies by using the density-functional theory. The NW has been constructed along the  direction. It was found that Ga vacancies prior to reside on the NW surface due to the lower formation energy. Interestingly, it was found that the GaN NWs with a Ga vacancy can show induced local magnetic moment in N atoms, with a magnetic moment about 0.60 uB. The study of vacancy-vacancy interactions indicates that the NWs prefer ferromagnetic ground state. We further confirmed that the total magnetic moments can be increased by increasing the Ga vacancy concentration without significant energy cost. The theoretical results are in good agreement with the recent experimental observations. Our findings are helpful to gain a more novel understanding of structural and spin properties of Ga vacancy in wurtzite GaN NWs, and also provide a possible way to generate the low dimensional magnetic GaN nanostructures by introducing Ga vacancies instead of doping with transition-metal atoms.