I assume you're asking whether a moving objects curves spacetime differently than a stationary one as its velocity increases.
Strictly speaking, no: it's the same spacetime geometry either way. The spaceship warps spacetime either way, and all we'd be is talking about it in a different frame. Because of this difference of frames, in a sense the gravitational field is different even though the geometry is the same.
If this spaceship, according to special relativity, gains mass as a factor of y as it approaches c, then its gravitational field should increase in strength as well.
This is not quite right. First, the spaceship does not gain mass. It's true that quantity $\gamma m$ is sometimes called relativistic mass, but this term is redundant with energy, bad at its intended purpose of preserving a superficial resemblance to Newtonian mechanics, and depreciated in physics. In special relativity, mass is invariant: $(mc^2)^2 = E^2 - (pc)^2$ is the same in all inertial frames.
Which is just as well, since the 'gravitational charge' isn't mass, but energy. But it is not a simple proportional increase when we view the spaceship's gravitational field in a frame in which it has a lot of energy.
This shouldn't be surprising if you know a bit of electromagnetism. A moving charge produces an electromagnetic field that has both electric and magnetic parts, since the motion of the charge, i.e. the current, matters. The electric field is enhanced in directions perpendicular to the direction of motion, which we can picture as the initially spherically symmetric field lines getting Lorentz-contracted, thus 'squishing' them close together in the perpendicular directions.
The gravitational analogue of electric current would be momentum, but because gravity is spin-2, stress in addition to energy density and momentum density is relevant to how spacetime is bent. You can see this described in the stress-energy tensor. So the gravitational field is more complicated, but it has an analogous behavior of being strengthened perpendicular to the direction of motion, although its quantitative behavior is different.
In the limit of lightspeed, the electromagnetic field of an electric charge turns into an impulsive plane wave, and the gravitational field of a point-mass behaves analogously, turning into a vacuum impulsive gravitational (pp-)wave.