No. The solutions to the relativistic version of the Kepler problem are different from the Newtonian ones:

You still have hyperbolic-like, parabolic-like and elliptic-like solutions. However, the parabolic ones can loop around the black hole a couple of times before going off to infinity, and the elliptic ones will have their perihelion precess around it. Additionally, you get solutions that cross the horizon and never come back out, eventually hitting the singularity, and trajectories that asymptotically approach a circular orbit.