  We consider the Dirichlet problem for the Schr\"odinger-H\'enon system $$
-\Delta u + \mu_1 u = |x|^{\alpha}\partial_u F(u,v),\quad \qquad
  -\Delta v + \mu_2 v = |x|^{\alpha}\partial_v F(u,v) $$ in the unit ball
$\Omega \subset \mathbb{R}^N, N\geq 2$, where $\alpha>-1$ is a parameter and
$F: \mathbb{R}^2 \to \mathbb{R}$ is a $p$-homogeneous $C^2$-function for some
$p>2$ with $F(u,v)>0$ for $(u,v) \not = (0,0)$. We show that, as $\alpha \to
\infty$, the Morse index of nontrivial radial solutions of this problem
(positive or sign-changing) tends to infinity. This result is new even for the
corresponding scalar H\'enon equation and extends a previous result by Moreira
dos Santos and Pacella for the case $N=2$. In particular, the result implies
symmetry breaking for ground state solutions, but also for other solutions
obtained by an $\alpha$-independent variational minimax principle.
