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Single-crystal diamond optomechanical devices have the potential to enable fundamental studies and technologies coupling mechanical vibrations to both light and electronic quantum systems. Here, we demonstrate a single-crystal diamond optomechanical system and show that it allows excitation of diamond mechanical resonances into self-oscillations with amplitude >200  nm. The resulting internal stress field is predicted to allow driving of electron spin transitions of diamond nitrogen-vacancy centers. The mechanical resonances have a quality factor >7×10^{5} and can be tuned via nonlinear frequency renormalization, while the optomechanical interface has a 150 nm bandwidth and 9.5  fm/sqrt[Hz] sensitivity. In combination, these features make this system a promising platform for interfacing light, nanomechanics, and electron spins.