We calculate the density-dependent spin-isospin asymmetry energy $J({k}_{f})$ of nuclear matter in the three-loop approximation of chiral perturbation theory. The interaction contributions to $J({k}_{f})$ originate from one-pion exchange, iterated one-pion exchange, and irreducible two-pion exchange with no, single, and double virtual \ensuremath{\Delta}-isobar excitation. We find that the approximation to $1\ensuremath{\pi}$-exchange and iterated $1\ensuremath{\pi}$-exchange terms (which leads already to a good nuclear matter equation of state by adjusting an emerging contact term) is spin-isospin stable, since $J({k}_{f0})\ensuremath{\simeq}24\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}>0$. The inclusion of the chiral $\ensuremath{\pi}N\ensuremath{\Delta}$ dynamics, necessary in order to guarantee the spin stability of nuclear matter, keeps this property intact. The corresponding spin-isospin asymmetry energy $J({k}_{f})$ stays positive even for extreme values of an undetermined short-distance parameter ${J}_{5}$ (whose possible range we estimate from realistic NN potentials). The largest positive contribution to $J({k}_{f})$ (a term linear in density) comes from a two-body contact term with its strength fitted to the empirical nuclear matter saturation point.