@Rennie is right. You are enunciating a conclusion that Mach's principle would loosely imply. Mach's principle is that it is the total of all stars (and he probably said or meant all the matter) in the universe that defines a reference frame in the universe, and that defines an inertial frame. If the stars (or matter) were rotating, then we'd be in an inertial frame if we were also rotating with it. Mach's principle was stated and argued most in relation to how a rotating reference frame could be inertial.
Mach's principle influenced Einstein to get to general relativity, or perhaps the idea that it was the total matter, even very distant matter, that defined a reference frame, or for Einstein, the spacetime. But General Relativity was not specifically made to abide by it. General Relativity does not conform with Mach's principle, and the principle was never really defined exactly in mathematical terms. In fact, in the Wikipedia article on it there are 10 different not equivalent statements of Mach's principle.
Einstein found some effects that were Machian, such as the Lens Thirring effect (see the wiki also) where an inertial reference frame inside a rotating shell processes with it. In General Relativity (GR) the rotating Black Holes (BH) such as the Kerr solution also has it that outside the horizon, but near it, particles, even light will rotate with it. This is called frame dragging in GR. But the effect diminishes and goes away further from the BH. In fact, this spacetime geometry 'rotation' can be used to extract energy from the BH's angular momentum, till it stops rotating (this was found by Penrose).
But the earth's rotation, as far as GR is concerned, you can have a rotating body like the Kerr BH and still have nothing else in the spacetime, nor in asymptotic infinity. And the rest of spacetime far from the rotating object is effectively not affected.
The understanding nowadays is that Mach had an intuitive contribution to thinking about inertia being influenced by the matter in the spacetime, but no firm predictable effect. You still have to solve the GR equations.
So no, the earth would still rotate, angular momentum is still conserved at the weak gravitational limit if the rest of the matter in the universe (and let us throw in the energy and dark energy also, and the universe would then be flat and not expanding) was not there. At least according to GR. Nobody has tested it, of course, but it is that way in GR.
What might be more interesting would be if you have all the other matter-energy rotate around in the universe, I'm guessing our inertial reference frame would be dragged. Distances may play a role on how much.