  We calculate cross sections and transverse momentum distributions for the
incoherent diffractive production of vector mesons $J/\psi$ and $\Upsilon$ on
heavy nuclei. In distinction to coherent diffraction, the nucleus is allowed to
break up, but except for the vector meson no new particles are produced in the
reaction. Within the color dipole approach, we derive the multiple scattering
expansion of the incoherent diffractive cross section as an expansion over
quasielastic scatterings of the color dipole. We also compare our results to
the measurement of the ALICE collaboration for incoherent $J/\psi$ production
at $\sqrt{s_{\rm NN}} = 2.76 \, \rm {TeV}$ and show predictions at
$\sqrt{s_{\rm NN}} = 5.02 \, \rm{TeV}$. We also briefly discuss a possible
contribution to $J/\psi$ production in peripheral collisions in the $70 \div 90
\%$ centrality class.
