General Relativity is not the process of replacing mass with relativistic mass in Newton's equations of gravity. The gravity of a moving body isn't even purely attractive.
Obviously, the overall magnitude of the effect of gravity cannot be much different for a moving body than for a stationary one, since "moving" is completely dependent on reference frame. Any body can be said to be in motion at nearly c, or at rest, depending on that choice of frame.
The answer is yes, the gravity of a moving body is different from that of a stationary one, but not necessarily "greater," not isotropic (not the same in every direction around the body) and not entirely attractive. It depends on neither the rest mass nor the "relativistic mass," but only on the Stress-Energy Tensor, of which the rest mass is only one component, and the RM doesn't appear in at all.
Oldprof continues to incorrectly state that the acceleration of a body is F/M, with M the relativistic mass, which is completely wrong. Oldprof, cut it out. Until you learn some relativity yourself, kindly stop misleading others.