[theThinker (OP)]

Let's say we have a space lab deep in outer space freely floating with no rotation about the fixed stars. It would be an ideal inertial frame. Let's perform Newton's globes experiment where two equal globes are connected by a cord and rotates freely about the CM center of the cord. From Newtonian mechanics, it is predicted that there will be a tension in the cord.

My question is whether this has somehow been confirmed by experiments in our space programs. Or are there related experiments that could provide indirect proofs for the experiment.

If no such experiment has been done, how much confidence should we put in our prediction that there is tension in the cord.   

[yor_on]

What you're thinking of is Mach universe right? And whether that is correct.

[yor_on]

then again, assuming that it is constants, and their relations, that defines your observation?

[yor_on]

Take it away please, you got two.

[Colin2B]

... how much confidence should we put in our prediction that there is tension in the cord.   

A great deal.
Can you give any reasons why there might not be?

[Ethos_]

If no such experiment has been done, how much confidence should we put in our prediction that there is tension in the cord.   

This experiment would only be valid if one were to also consider the combined mass of both globes and the velocity of their rotation. If the gravitational attraction between both globes were sufficient, it could overcome the centrifugal force created by their rotation. The tension that might develop all depends on their combined mass and speed of rotation, and that is a physical fact born out by experiments we can preform here on earth.

[theThinker (OP)]

... how much confidence should we put in our prediction that there is tension in the cord.   

A great deal.
Can you give any reasons why there might not be?

I think so too. There seems to be no reason why Newtonian mechanics should suddenly fail.

The reason I ask is because of these many disputes of Mach against the "absolute" space of Newton and the talk about the effect of the mass distributions of the fixed stars.

What if there is an absolute space of Newton's? The tension depends only on rotation in such a space. 

[Colin2B]

The reason I ask is because of these many disputes of Mach against the "absolute" space of Newton and the talk about the effect of the mass distributions of the fixed stars.What if there is an absolute space of Newton's? The tension depends only on rotation in such a space. 

Things have moved on from Mach's time. We now know that the fixed stars are not fixed. Any mass in the universe will theoretically affect an area of space - as Ethos pointed out, even the 2 spheres will influence each other.
Although you talk about the spheres being in an inertial frame, the spheres themselves are not moving in an inertial frame because they are not moving in an inertial manner, they are constantly accelerating, being pulled out of a straight path by the string. This is what causes the tension on the string and it would take more than the influence of distant stars to cause significant deviation - as you can prove by performing the experiment on earth.
Mach's observation was that you can't consider the spheres to be at rest and the stars to be rotating around them because there would be no tension in the string, but as I said the spheres are not an inertial frame when rotating so there isn't really an issue about absolute frames. It's worth reflecting that if you were inside one of the spheres you would be pushed to one side as if by gravity, so Einstein's equivalence principle would still apply.

[rmolnav]

In that scenario, supposed with no gravitational pull from distant stars, we should take into consideration that globes pull each other too.
Depending on their masses and separation, they could keep rotating (after somehow initiating the movement) even without any cord, at a certain low speed, similarly to rotating twin stars, but at a very, very tiny scale.
Beyond that velocity, the cord would be necessary to maintain the distance ...
If not exactly that, equivalent experiments must have been carried out in ISS, I guess.