Newtonian physics does have the Doppler effect which can cause red shift, but it doesn't accurately explain the red shift that occurs when observing distant galaxies. That's because this red shift occurs due to three factors:

1) The galaxy is moving away from us. This is most like the classic Doppler effect, but because of the high relative velocities involved, time dilation needs to be taken into account to model the red shift accurately, thus at least special relativity should be used.

2) The light travels through space with different curvature. For example, light originating near a very massive star will red shift when moving away from that star because it moves into less curved space. General relativity is needed to explain this effect.

3) The light travels through expanding space. For very distant galaxies this becomes the dominating factor of red shift, as we see an amount of red shift directly corresponding to their distance from earth. General relativity also explains this effect.

I see, thank you.

Yeah, you’re correct, you don’t need to invoke relativity to explain the Doppler effect. Sibling comment did a good job explaining the ways relativity does impact redshift, but my initial statement (“because of relativity…”) was not correct.