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We analyze the stability of spin spiral states in the two-dimensional Heisenberg model. Our analysis reveals that the SU(2) symmetric point hosts a dynamic instability that is enabled by the existence of energetically favorable transverse deformations---both in real and spin space---of the spiral order. The instability is universal in the sense that it applies to systems with any spin number, spiral wavevector, and spiral amplitude. Unlike the Landau or modulational instabilities which require impurities or periodic potential modulation of an optical lattice, quantum fluctuations alone are sufficient to trigger the transverse instability. We analytically find the most unstable mode and its growth rate, and compare our analysis with phase space methods. By adding an easy plane exchange coupling that reduces the Hamiltonian symmetry from SU(2) to U(1), the stability boundary is shown to continuously interpolate between the modulational instability and the transverse instability. This suggests that the transverse instability is an important mechanism that hinders the formation of a spin superfluid, even in the presence of strong exchange anisotropy.