In Newtonian mechanics the future is predictable, If we know the initial state of a system exactly, then the laws of physics determine its state arbitrarily far into the future.
One has to separate the mathematical predictability, i.e. the ability to give a mathematical formula that predicts a situation, and validity i.e. is this a valid prediction and what nature will do in the future?
The answer to the second is that Newtonian mechanics predictions are already invalidated in the earth-satellite system. For GPS to work one has to use both special and General realtivity, even for the very near future.
What if the usual predictability of the laws of physics fail in general relativity
Physicists will have to define new models to fit and predict in this new regime , if necessary.