This is one of those cases where somebody stumbles upon an actually interesting problem, with some deep and important physics behind it, but then totally and completely screws up the answer.
It's absolutely true that Newtonian gravity (which propagates instantly) will give you stable orbits for binary pairs. Furthermore, it's even true that if the propagation speed for gravity is not infinite, as in General Relativity, you cannot have stable orbiting pairs. But none of the rest of the argument follows. The gravitational fields of moving bodies are not simple. You can't get them by just looking at the time-delayed position of where the object was. This isn't controversial either, the exact same effect is readily observable with moving electric charges, and leads directly to magnetism. Similar stuff happens with gravity. But if you do the calculations right (and they haven't even come close to doing so), the orbits aren't unstable because they spiral outwards, they're unstable because they spiral inwards.
And again, these effects are observable with electric charges (where we know the propagation speed of the field quite well). They're even observable with gravity. Binary orbits decay. They shouldn't, if Newtonian gravity was correct. But it isn't, so they do.
None of this is controversial. None of it is even mysterious. But it does require actually knowing more than high school physics. And the author obviously doesn't.