The dissociation dynamics of a dichromatically laser-driven diatomic Morse molecule vibrating in the ground state is investigated by applying tools of the nonlinear theory of classical Hamiltonian systems. Emphasis is placed on the role of the relative phase of the two fields, phi. First, it is found that, just like in quantum mechanics, there is dependence of the dissociation probability on phi. Then, it is demonstrated that addition of the second laser leads to suppression of probability (stabilization), when the intensity of the first laser is kept constant just above or below the single laser dissociation threshold. This "chemical bond hardening" diminishes as phi increases. These effects are investigated and interpreted in terms of modifications in phase space topology. Variations of phi as well as of the intensity of the second laser may cause (i) appearance/disappearance of the stability island corresponding to the common resonance with the lowest energy and (ii) deformation and movement of the region of Kolmogorov-Arnold-Moser tori that survive from the undriven system. The latter is the main origin in phase space of stabilization and phi dependence. Finally, it is shown that the use of short laser pulses enhances both effects.