It's not infinitely think or even half of an infinity thick because the heat transfer to an infinite wall of ice would have frozen us and I'd be very cold right now, which I'm not.
If the ice wall exists it's certainly very thin and fragile and we should all worry.
You don't just need a large thermal mass, you need a large rate of heat transfer.
Ice is a good insulator.
If it was just a matter of how much, all the ice would have melted by now.
Instead it acts kind of like a warm house in the middle of a freezing winter, with the heater keeping the house warm but a constant output of heat, which escapes through the walls.
If we were in an infinite mass plaen, not above, but in plane since the walls are walls, would the lateral gravity of it all crush us, pull us appart, be net zero?
That math is hard.
What I can tell you is that there will be a force going outwards towards the ring.
What I can't tell you for certain is if that force is finite or infinite.
According to demos, the force will not be infinite, at least for some dimensions.
https://www.desmos.com/calculator/rklalgzmexIf you have the wall infinitely short, with you in the same plane as the wall, and ignore the constant terms (G, your mass, the area density of the wall, the hypothetical planar Earth beneath you), and take your distance from the centre to be r, then if the inner side of the wall is at 2r, the force will be ~1.746.
If the inner side of the wall is 1.001*r, the force will be ~13.98.
In both cases that is towards the wall closest to you.
The extension would be what would happen to the ring. As the force is finite, I think it should hold itself together.
While each ring would have a force pulling it outwards from all the further out rings, they would also have a force pulling them inwards from the inner rings, meaning no where should get an infinite pressure.
But it might be enough to overcome the yield strength of the ice/Earth, at which point it would be torn apart and crushed into an infinitely large ring.