The energies and matter densities of finite nuclei under radial compression are predicted by using a constrained Hartree-Fock method with the $\ensuremath{\Delta}$ degree of freedom included. The results are presented for ${}^{90}$Zr in a calculation within a model space of seven major oscillator shells. The main feature of this calculation is that the effective baryon-baryon interactions associated with the $\ensuremath{\Delta}$ are evaluated within a $G$-matrix approach based on a coupled-channel $\mathrm{NN}\ensuremath{\bigoplus}N\ensuremath{\Delta}\ensuremath{\bigoplus}\ensuremath{\pi}\mathrm{NN}$ model that can describe the $\mathrm{NN}$ data up to 1 GeV. It is found that as the nucleus is compressed to about 2--3 times of the ordinary nuclear density, the $\ensuremath{\Delta}$ component is sharply increased to about 10% of all baryons in the system. This result is consistent with the values extracted from relativistic heavy-ion collisions.