Can two heavy objects circling around their C.M. be separated because of the speed of gravity? Imagine two massive objects, with the same mass (M) circling around their center of mass (C.M.). Let's assume that the distance between them is 1 light hour. Don´t the two bodies get accelerated and move away from each other because they feel the gravity of each other as it was 1 hour ago, because of which a force tangent to the direction of the speed develops?
Can it be that gravity acts instantaneously? I've read about experiments proving the speed of gravity to be equal to the speed of light, but these were disputed.
Isn't there both in Newtonian mechanics and GR there a time delay? The difference being though that space and time in Newtonian mechanics are separate and absolute (implying instantaneity?), while in GR the both are a whole and absolute.
Can two heavy objects circling around their C.M. be separated because of the speed of gravity?

No. Newtonian mechanics does quite well assuming instantaneous gravity. You can get relativistically correct orbits either by doing a complete GR calculation, or using a simple approximation as already eluded alluded to in @antispinward's answer.
This excellent answer to Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch? in Space Exploration SE explains that we get the wrong answer if purely Newtonian mechanics is used except for slowing down the speed of gravity, but that's because one is neither treating gravity correctly nor using Newtonian mechanics correctly.
Many answers to How to calculate the planets and moons beyond Newtons's gravitational force? include a way to treat this problem using a well-accepted approximate treatment of general relativity.
From this answer:

The acceleration of a body in the gravitation field of another body of standard gravitational parameter $GM$ can be written:
$$\mathbf{a_{Newton}} = -GM \frac{\mathbf{r}}{|r|^3},$$
where $r$ is the vector from the body $M$ to the body who's acceleration is being calculated. Remember that in Newtonian mechanics the acceleration of each body depends only on the mass of the other body, even though the force depends on both masses, because the first mass cancels out by $a=F/m$.

and later:

The following approximation should be added to the Newtonian term:
$$\mathbf{a_{GR}} = GM \frac{1}{c^2 |r|^3}\left(4 GM \frac{\mathbf{r}}{|r|} - (\mathbf{v} \cdot \mathbf{v}) \mathbf{r} + 4 (\mathbf{r} \cdot \mathbf{v}) \mathbf{v} \right),$$