In the case where the object is moving slowly or not near a VERY large gravity field Newtonain mechanics and relativity give EXACTLY the same results.
Am I wrong in stating that they DO NOT give the exactly same results, only that the differences are too minute for our ability to measure them?
True there would be an effect, but it would be so small that in every practical way you can and should ignore it. For an object moving at 30meters/ second the differece would be 1/sqrt(1 – 10^14) which is in effect 1, IE no difference at all.