Using infinite projected entangled-pair states, exact diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg model undergoes a spontaneous dimerization on the square lattice, in contrast with its SU(2) and SU(3) counterparts, which develop Neel and three-sublattice stripelike long-range order. Since the ground state of a dimer is not a singlet for SU(4) but a 6-dimensional irreducible representation, this leaves the door open for further symmetry breaking. We provide evidence that, unlike in SU(4) ladders, where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is additionally broken, leading to a gapless spectrum in spite of the broken translational symmetry.