By means of quantum-dynamical and classical trajectory calculations of H₂⁺ photodissociation in strong laser fields, it is shown that for certain combinations of pulse durations and intensities the rotational dynamics can lead to fragmentation. In that case, the photofragments exhibit characteristic angular distributions. The classical calculations provide a transparent physical picture of this mechanism which is also very well established in collisions between atomic nuclei or liquid droplets: non-rotating systems are stable, whereas rotating systems fragment due to the decrease of the fragmentation barrier with increasing angular momentum.