Not exactly!
The point is that the "force" of gravitation between two objects is toward their instantaneous position or very nearly. This for Newton led to instantaneous action at a distance. Einstein's solution is a bit more complex, but essentially it has two parts. The first is that the gravitational field of an "inertial" object essentially moves with it, rather than lagging behind. This allows the force between two objects to act toward their instantaneous position. If gravity did not work this way planetary orbits would not be stable.
The speed of light limit comes in when considering changes to the field and only when considering changes. A change in mass or an acceleration of one or more of the involved objects. One thing to keep in mind is that in this limited case an orbit is not an accelerating frame of reference.., gravitationally. It is uniform.., the planet is in free fall.
Newton needed an instantaneous action at a distance because from his perspective gravity is a force acting between two objects. Einstein changed that by describing gravity as a field, manefest as a curvature of space. Since objects don't just appear or disappear and the redistribution of mass even during catastrophic events happens at far less than the speed of light, changes to an established gravitational field are subtle and slow, when compared to c.
This moves a bit off the mainstream view but, since the gravitational field is a curvature of space and space is dynamically interacting with matter, the field moves with an object, as long as the motion is uniform and constant. Frame-dragging, even though we must consider it to be a very weakly defined interaction, must over time have a cumulative effect that results in space being a dynamic component of any large scale inertial system.., a solar system or even a galaxy...
