No, the classical Doppler effect and the Galilean principle of relativity do not conflict with each other. The classical Doppler effect describes the perceived change in frequency of a wave (such as sound or light) when there is relative motion between the source of the wave and the observer.
According to the Galilean principle of relativity, the laws of physics should be the same in all inertial reference frames. An inertial reference frame is a frame of reference that is not accelerating. In this context, "no mechanical experiment is able to tell two inertial frames apart" means that the laws of physics, including the principles governing waves, should be the same for observers in different inertial frames.
When applying the Galilean principle of relativity to the classical Doppler effect, the relative motion between the observer and the source of the wave is accounted for by the transformation of coordinates and velocities between the two frames of reference. The observed frequency shift due to relative motion can be explained consistently within the framework of the Galilean principle of relativity.
However, it's important to note that the Galilean principle of relativity was superseded by Einstein's theory of special relativity. In special relativity, the principle of relativity is extended to include the constancy of the speed of light in all inertial frames. This leads to the relativistic Doppler effect, which differs from the classical Doppler effect at high velocities. In the relativistic Doppler effect, both time dilation and the Lorentz contraction of space come into play, resulting in more complex frequency shifts.
In summary, the classical Doppler effect and the Galilean principle of relativity are compatible within the framework of classical physics. However, when considering high velocities or phenomena involving electromagnetic waves, the relativistic Doppler effect and the principles of special relativity become relevant.