No, classical general relativity is entirely time symmetric: time symmetry means here that if you have a solution for the equations that describe nature, then y

No, classical general relativity is entirely time symmetric: time symmetry means here that if you have a solution for the equations that describe nature, then you still have a solution after you make the coordinate transformation t -> -t. This follows directly from the fact that the Einstein field equations involve only tensors and are thus diffeomorphism invariant. For the case of a black hole, the solution you get by reversing time describes a white hole., which is again an entirely valid solution to the Einstein field equations.