From the context of non-relativistic classical mechanics, Newton's bucket argument says that angular velocity is absolute, just as Newton's first law says that translational acceleration is absolute.
The modern view of Newton's first law is that it says there exists a set of preferred frames of reference (frames in which the laws of physics take on their simplest form) called inertial frames. In classical mechanics, all inertial frames have two things in common: That the origin of one inertial frame as viewed from the perspective of some other inertial frame is moving at a constant velocity, and that the axes of any two inertial frames are not rotating with respect to one another.
What this means is that any two inertial observers will agree on the translational acceleration and angular velocity of any object. Another way to say this is that translational acceleration and angular velocity are absolute in non-relativistic classical mechanics.
Things get a bit trickier from the perspective of general relativity. General relativity preserves the basic concept of an inertial frame, but inertial frames in general relativity are not the same as those in non-relativistic classical mechanics. Inertial frames in general relativity are local rather than universal, and a non-rotating frame centered on a free-falling object is inertial in general relativity (but not in Newtonian mechanics).
Nonetheless, angular velocity and acceleration still have a bit of absoluteness to them in general relativity. One can construct local experiments that measure proper angular velocity and proper translational acceleration. A modern smartphone contains (low quality) versions of such devices, a MEMS gyro and a MEMS accelerometer.