It's curved
spacetime which does the job. In fact, for the planets of the solar system, the temporal part of the curvature (gravitational time dilation) is more important in determining orbits. Only rapidly-moving objects sample the space curvature to any great extent: hence the deviation from Newton we see in the orbit of Mercury, for instance.
Or so says Bernard Schutz, director of the division of Astrophysical Relativity at the Max Planck Institute, in his book
Gravity: From The Ground Up. His Chapter 17 on spacetime geometry has two successive sections entitled
Newtonian gravity as the curvature of time and
Do the planets follow the geodesics of this time-curvature?. In these two sections he uses a metric which has Schwarzschild's time coordinates in flat space, and demonstrates that it generates a Newtonian gravitational redshift and Newtonian orbits.
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We have therefore found a curved-spacetime picture of Newtonian gravity. The curvature here is only in the time-direction. Curvature in time is nothing more than the gravitational redshift: time advances at different rates in different places, so time is curved. We have found that the gravitational redshift fully determines the trajectories of particles in the gravitational field.
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As to whether spacetime curvature or gravitational force holds the planets in their orbits: under GR it's the spacetime curvature which modifies the path of a freefalling object into a closed curve; the object accelerates
as if subjected to a force. Whether you can say that spacetime curvature is what
really causes the deviation from straight-line movement is a philosophical point that I think has been debated on BAUT quite recently.
Grant Hutchison