As a way to find analogies and differences in the dynamics of hot and dense matter under extreme conditions, we present the first self-consistent relativistic-hydrodynamic calculations of both neutron-star mergers and lowenergy heavy-ion collisions employing the same equation of state. By a direct comparison of the evolution of quantities such as temperature, entropy, and density, we show that neutron-star collision regimes can be probed directly at GSI beam energies. We provide concrete evidence that the physical conditions reached in binary neutron-star mergers can be studied in present and future laboratory experiments, thus bridging 18 orders of magnitude in length scale, from microscopic ion collisions to macroscopic astrophysical compact objects.