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This would be possible if there was a material unaffected by gravity, right?
I think in that case, the earth would just depart the location of the train, leaving it drifting in space.
I was assuming the rails are strong enough to keep the train on the Earth, but I guess infinite friction from the movement and rotation of the Earth probably isn't survivable by any railway material. Hypothetically, if you had a material unaffected by gravity (train), and a material that is absolutely invincible (the rails, and they are anchored to the center of the Earth), now does it work?
It's impossible to even reason about what something entirely unaffected by gravity might or might not do because everything, including gravity, is relative. Something not affected by Earth's gravity would probably get carried away by the wind, but something not affected by the Sun's gravity would float away from the sun (relative to the Earth) and appear to have some sort of mysterious acceleration from our perspective. If the object isn't affected by the Milky Way's gravity, even more shenanigans.
No, the problem is not gravity, is that the train attached to earth has velocity dictated by the Earth movements, and keeps it because of inertia. In your theoretical experiment, the train would be launched on space at constant velocity.