In the upcoming decade cadenced wide-field imaging surveys will increase the number of identified $z<0.3$ Type~Ia supernovae (SNe~Ia) from the hundreds to the hundreds of thousands. The increase in the number density and solid-angle coverage of SNe~Ia, in parallel with improvements in the standardization of their absolute magnitudes, now make them competitive probes of the growth of structure and hence of gravity. The peculiar velocity power spectrum is sensitive to the growth index $gamma$, which captures the effect of gravity on the linear growth of structure through the relation $f=Omega_M^gamma$. We present the first projections for the precision in $gamma$ for a range of realistic SN peculiar-velocity survey scenarios. In the next decade the peculiar velocities of SNe~Ia in the local $z<0.3$ Universe will provide a measure of $gamma$ to $pm 0.01$ precision that can definitively distinguish between General Relativity and leading models of alternative gravity.
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