I am not an astrophysicist, but my understanding is that melting the ice would cause it to escape into space for at least two reasons: 1) Mars' core has cooled

I am not an astrophysicist, but my understanding is that melting the ice would cause it to escape into space for at least two reasons: 1) Mars' core has cooled significantly, and it no longer has a powerful magnetic field to protect it from solar winds. Solar winds have the ability to take a polarized molecule like H2O and strip it from the atmosphere. It's also much easier to strip it from Mars because 2) Mars has a gravity of approximately 1/3rd of what Earth has. Combined, these two factors mean that Mars has an atmosphere that is 0.6% as thick as Earth's.