How can gravity be independent from velocity? If gravity is directly related to mass, and mass increases as an object travels closer to light-speed, how can velocity not affect gravity?
 A: You are basically asking whether an object's gravitational field strength (causing spacetime curvature and time dilation) does increase with the relative velocity of the object.
You are confused because of the two definitions of mass, rest and relativistic.
As per SR, the rest mass of an object is independent of its relative velocity, it is the same for all observers in all reference frames.
Relativistic mass, is dependent upon the velocity of the observer.

According to the concept of mass–energy equivalence, the rest mass and relativistic mass are equivalent to the rest energy and total energy of the body, respectively.
  The measurable inertia and gravitational attraction of a body in a given frame of reference is determined by its relativistic mass, not merely its rest mass. For example, light has zero rest mass but contributes to the inertia (and weight in a gravitational field) of any system containing it.

https://en.wikipedia.org/wiki/Mass_in_special_relativity
A photon, having no rest mass, still does have stress-energy, gravitational potential, and does curve spacetime.
I have actually asked a question about this:
Do photons bend spacetime or not?
Now, contrary to popular belief, the gravitational potential (gravitational field strength) of an object depends on its stress-energy (not mass). 
The stress-energy of an object increases with relative velocity.
Thus, you would expect an object's gravitational field strength to depend on its relative velocity.

The metric tensor is invariant with respect to coordinate transformations, so if we take some observer moving at near light speed they would also find the gravity round the Sun to be described by the Schwarzschild metric. It will not look the same in the observer's coordinates, that is the individual components gij will be different, but it will be the same tensor. Since in the observer's frame they are stationary and the Sun is moving, the conclusion is that velocity does not change the spacetime curvature.
  Incidentally, this is why a fast moving object does not turn into a black hole.

Does a moving object curve space-time as its velocity increases?
It is very important to understand that the gravitational field strength, gravitational potential (that determines spacetime curvature caused by the object, and time dilation) and the metric tensor is invariant with respect to coordinate transformations, thus the gravitational field strength (causing spacetime curvature) of the object does not change with relative velocity.
