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The ab initio phase diagram determines the energetic stability of the icosahedral TiZrNi quasicrystal. The complete ab initio zero-temperature ternary phase diagram is constructed from the calculated energies of the elemental, binary and ternary Ti-Zr-Ni phases. For this, the icosahedral i-TiZrNi quasicrystal is approximated by periodic structures of up to 123 atoms/unit cell, based on a decorated-tiling model [R. G. Hennig, K. F. Kelton, A. E. Carlsson, and C. L. Henley, Phys. Rev. B 67, 134202 (2003)]. The approximant structures containing the 45-atom Bergman cluster are nearly degenerate in energy, and are all energetically stable against the competing phases. It is concluded that i-TiZrNi is a ground-state quasicrystal, as it is experimentally the low-temperature phase for its composition.