Actually, we don't use relativity so much. For some stuff like light time delays, yes, but for orbit determination we use newtonian mechanics still. It's just that once you start adding more than two bodies, you have to shift to doing numerical integration to find solutions, which doesn't work for real-time stuff like video games.

For most applications, F = -GM1M2/r^2 is still a valid approximation.

Edit: And by most applications, i mean sending a satellite to a planet in our solar system. If you were trying to go to Alpha Centauri, you damn sure better use Relativity to model your trajectory.