Group III nitrides are widely used in commercial visible-wavelength optoelectronic devices, but materials issues such as dislocations, composition fluctuations, and strain negatively impact their efficiency. Nitride nanostructures are a promising solution to overcome these issues and to improve device performance. We used first-principles calculations based on many-body perturbation theory to study the electronic and optical properties of small-diameter InN nanowires. We show that quantum confinement in 1 nm wide InN nanowires shifts optical emission to the visible range at green/cyan wavelengths and inverts the order of the top valence bands, leading to linearly polarized visible-light emission. Quantum confinement on this scale also leads to large exciton binding energies of 1.4 eV and electronic band gaps in excess of 3.7 eV. Our results indicate that strong quantum confinement in InN nanostructures is a promising approach to developing efficient visible-wavelength light emitters.