EdenOne:
Newtonian physics also does not take into account the fact that all motion is relative and that the speed of light is constant no matter what the observer is doing. If for example you were in a spacecraft travelling at the speed of light and shone a torch out of the front window then Newtonian physics would say that you would see a beam of light out of the window. Relativity states that the beam of light would travel at the speed of light, the same as your spacecraft and hence never leave the torch.
In Newtonian physics the answer would depend on ones view of light.
If one assume light travels trough a medium like the eather, then one would expect an observer could break the "light barrier" in which photons would not leave the torch. In this view the torch would be blue-shifted 2x when pointing backwards and not shining when pointing forwards.
If however one assumes light just consists of particles (as was also believed for a long time) the light would according to good old gallilean relativity just behave like it was standing still -- notice this is 100% consistent with the predictions of relativity. Newtonian physics is in this view consistent with relativity.
If however we consider an outside observer (or what the spaceman sees of the outside world) things become more tricky and one must keep in mind that the same explanation must go for both observers since, by relativity, one will see the universe hurling towards him and the other will see the spaceship travelling towards him near the speed of light, and both will see time travelling much slower for the other person (think about that for a moment!).
Long story short, the spaceman looking out of the window will see everything behind him very red-shifted, and things in the front of him very blue-shifted. He will also think (or rather, this will be true for him) that everything at the front or behind him is very near to his spaceship, i.e. that alpha-centauri is only a few kilometers away due to length contraction. When he shine his light, he will see the light leave the torch just as it should and travel a very short distance towards the star he is approaching. The person on the outside will see the spaceman turn on his terribly blue/red-shifted torch (depending on where he is) in slow-mo.
(updated)