Relativistic corrections of order ${\alpha }^2$ to the $g$ factor of ${\mathrm{H}}_2^{+}$ are calculated with a high accuracy of nine significant digits for a wide range of rovibrational states. The precision of previous calculations [R. A. Hegstrom, Phys. Rev. A 19, 17 (1979) is improved by about five orders of magnitude by performing nonadiabatic variational calculations and by including recoil corrections. These results allow for nondestructive identification of the internal state through the measurement of spin-flip transition frequencies, which is a crucial requirement for proposed spectroscopy experiments on ${\mathrm{H}}_2^{+}$ and its antimatter counterpart ${\overline{\mathrm{H}}}_2{}^{-}$ in Penning traps [E. G. Myers, Phys. Rev. A 98, 010101(R) (2018)]. Further, they pave the way towards precision calculations of the $g$ factor through the calculation of higher-order QED corrections and hence to an alternative precision route to obtaining the proton-electron mass ratio.