I have a feeling that while that would not be its official name, it would become a nickname for the system.
It's quite common for nicknames or marketing names to be used for systems. The TTC has the "Rocket" moniker for the subway system, London has "the Tube" while still technically being called the London Underground.
While I could see it used as the official name (much like Berlin's S-Bahn or Paris' RER), I could also see it used as the casual reference, because "GO Regional Express Rail" would be quite a mouthful.
Also, given the future classifications of GO between Electrified Regional Express Rail and All-Day Two-Way Diesel service, I can see the need for some kind of casual distinction between the two. Whether it's officially adopted by GO or merely used casually, I'm sure that something will pop up.