  We characterize the universal far-from-equilibrium dynamics of the
two-dimensional quantum Heisenberg magnet isolated from its environment. For a
broad range of initial conditions, we find a long-lived universal prethermal
regime characterized by self-similar behavior of spin-spin correlations. We
analytically derive the spatial-temporal scaling exponents and find excellent
agreement with numerics using phase space methods. The scaling exponents are
insensitive to the choice of initial conditions, which include coherent and
incoherent spin states with values of total magnetization and energy in a wide
range. Compared to previously-studied self-similar dynamics in non-equilibrium
$O(n)$ field theories and Bose gases, we find qualitatively distinct scaling
behavior originating from the presence of spin modes which remain gapless at
long times and which are protected by the global SU(2) symmetry. Our
predictions, which suggest a new non-equilibrium universality class, are
readily testable in ultra-cold atoms simulators of Heisenberg magnets.
