The Schrödinger equation isn't invariant under Galilean transformations. It's invariant under a central extension of the universal cover of the Galilean group. Specifically, this means that you cannot just apply a Galilean transformation to the wave-function, but you need to add a (mass and time dependent) phase factor in order to transform a solution into another solution. In contrast to the Lorentz group, the phase factor cannot be made to vanish in case of the Galilean group.
I want to prove that the time dependent Schrödinger equation is invariant under Galilean transformation.
See chapter 3 Ballentine - QM - A Modern Development - he does the converse - proves the Schrodinger equation from the Galilean principle of relativity. Specifically he assumes the probabilities of an observation are invariant.
BTW I think Rubi is right although I haven't gone deep into it.