  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 $\Delta$-isobar excitation. We find that the approximation
to $1\pi$-exchange and iterated $1\pi$-exchange terms (which leads already to a
good nuclear matter equation of state by adjusting an emerging contact-term) is
spin-isospin stable, since $J(k_{f0})\simeq 24 {\rm MeV}>0$. The inclusion of
the chiral $\pi N\Delta$-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.
