If I am aboard a spacecraft and have with me two identical metal spheres having rest mass $= m$, what happens to the gravitational force between these two spheres as the space craft velocity approaches the speed of light. According to Newton, $F = G*m1*m2/r2$. Does $F$ between my two spheres increase according to this equation as the relativistic mass increases, or is it only a function of the rest mass of the two spheres?
In the rest mass system of your spacecraft, there will be no difference in measusring the force of one sphere on the other.
A good example is that Newtonian celestial mechanics is very successful in describing the planetary system. There exists a star in our galaxy that is moving with 8% of the velocity of light, but at the rest frame of our star, this makes no difference.With respect to the rest system of that star we are moving at 8% of the velociy of light. Observers on that star's reference system would see our planetary system distorted and would need to unscramble newtonian gravity's $1/r^2$ in our rest system .
Relativistic mass is not a useful concept, except if one wants to calculate the amount of fuel needed for a craft to reach a specific velocity with respect to a given star.
The relativistic force transformation answers your question regardless of whether the force is gravitational, mechanical, or electrical. Therefore, if you place the masses a distance $d$ apart from each other perpendicular to the direction of velocity $v$, the $g$-force should reduce by the inverse of the traditional gamma factor as measured by the lab observer located outside the craft. However, if you locate the masses in a way that $d$ is parallel to $v$, the $g$-force remains unchanged (measured the same as in the craft's rest frame) from the viewpoint of the lab observer.
I do not know how exactly GR demonstrates the forces in your example, however, they must comply with the results predicted by SR while the masses are not accelerated very much due to the mentioned $g$-force nor the field affects the uniformity of the spacecraft motion at a constant $v$.