There is a property of spacetime which is independent of frame of reference.
The geometrical properties of the spacetime are described by the metric tensor, $ \eta _{\alpha \beta} =diag({-1,1,1,1})$ is SR (flat spacetime) or more generally $ g_{\alpha \beta} $ (any spacetime) in GR. This tensor specifies the distance between two infinitesimally close spacetime events, for example $ (t_1,x_1,y_1,z_1) $ and $ (t_1+dt,x_2+dx,y_2+dy,z_2+dz) $ in cartesian coordinates. The representation of this tensor is depending on your your choise of coordinates, but it describes the same physical (more accurate - geometrical) object. With the metric tensor you can calculate proper distances (called lorentz invariants in SR) between any two spacetime coordinates, quantities which are invariant under any coordinate transformation (= independent of frame of reference you're using, not necessarily inertial).
First off: as you mention: what is known as 'Mach's principle' was proposed by Einstein. (Also: Mach's austere philosophy of science is opposed to such a grand statement.) Einstein coined the name 'Mach's principle'. I want to emphasize that it was proposed by Einstein so I call it 'Einstein's Mach's principle.
Historians of physics describe the following:
Around 1915 Einstein was convinced that Einstein's Mach's principle is fundamental to GR.
Among the logical implications of the form of Einstein's Mach's principle that Einstein used was that in order for spacetime to exist there must be distribution of matter/energy in it.
A couple of years later the dutch astronomer Willem de Sitter found a solution to the Einstein Field Equations that describes a Universe with no matter in it.
This result was opposite to Einstein's expectation. Historians of physics describe that for quite a while Einstein tried hard to show that there was some mathematical error. Einstein worked to show that the de Sitter solution was actually invalid. But in the end Einstein had to admit that the de Sitter solution was in fact valid.
Historians of science describe that after that Einstein ceased to mention Einstein's Mach's Principle in articles about GR.
In 1954, in a letter written in reply to a specific question Einstein wrote: "Von dem Machschen Prinzip sollte man eigentlich überhaupt nicht mehr sprechen." (We shouldn't talk about Mach's principle anymore, really.)
(Written communication between Einstein and Felix Pirani)
Main source for this information:
Michel Jansen, 2008, Einstein's Quest for General Relativity, 1907-1920
Additional reading:
John Norton, Mach's principle before Einstein
My understanding is that there are multiple versions of "Mach's principle" in circulation. It seems to me that in order to begin to assess the question first the multiple versions of the principle would need to be categorized.
Einstein's version
I don't know which version the principle was in Einstein's thoughts, but it is interesting to see that Einstein's version was rendered untenable by the fact that de Sitter's solution is valid.
Decades later Johh Wheeler coined the phrase: "Curved spacetime is telling inertial mass how to move, inertial mass is telling spacetime how to curve." (Or words to that effect.) It appears to me that Einstein held to view where inertial mass is not only telling spacetime how to curve, but that in order for the Einstein Field to exist presence of inertial mass is necessary. (I use the expression 'Einstein Field' here as meaning: that which is described by the Einstein Field equations. The Einstein Field equations describe Einstein spacetime.)
It would appear that Einstein held a view (prior to being confronted by the de Sitter solution) that the existence of spacetime and inertial-mass-in-spacetime is fundamentally co-dependent. It would appear that this concept of co-dependency was sufficient to satisfy the version of Einstein's Mach's principle that Einstein used.
Best Answer
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.