Starting from the triple point, is the melting line between solid-phase and liquid-phase infinite? If not, why does it end? Because pressures are so high that classical inter-molecular interactions break down? Is this related in any way to the concept of degenerate matter? Can such pressure (at which the line ends) be easily computed?
No, the boundary doesn't suddenly "end" or "fade away", as the liquid-gas boundary fades away near the critical point.
Instead, the sudden end indicates that many other things may happen in the region of these extremely high pressures and the diagram doesn't want to discuss those because they're outside the limits of interest of the author of the diagram. By other things, I mean primarily new triple junctions that separate new phases of "ice", like those on this more detailed phase diagram of water:
Fifteen phases of ice are known today and the other triple junctions are counted as "triple points", too.
The required pressures to see these new phase transitions are extreme – thousands of atmospheres or much more – but the new phases are still much less extreme than the matter of white dwarfs or neutron stars. It's still some kind of ice that just wants to arrange a bit differently than normal ice.