Gravity is generated by energy, and mass is just one type of energy. For example photons produce a gravitational field of their own, and are influenced by gravitational fields.
So yes, energy in the form of velocity does produce gravity.
It can get complicated to calculate though, because velocity in the same direction as the observer doesn't exist, only in other directions (including perpendicular).
A thought experiment: Two huge identical masses are traveling toward each other, and perpendicular to an observer. They collide and "stick" to each other. All their velocity is now rest energy, which obviously has gravity.
But this means that even before they collided they also had gravity (since you can never create or destroy gravity, only move it).
Now rotate the experiment - one mass is traveling toward the observer, and the other is traveling with the observer. This time when they collide part of it is rest energy, and the rest is kinetic (since the observer is still moving, while the two objects stopped each other dead still at a point in space).
But this time, the mass that was with the observer had zero gravity (from the POV of the observer), but the mass traveling toward the observer had more gravity.
But it means that gravity is not a "global" phenomenon, it depends on your POV. And that makes for some difficult calculations if you must account for multiple objects all moving. (For example in cosmology. From my understanding the calculations are so complicated that they are mostly just ignored because few objects are moving fast enough for it to matter.)
Thank you for the answer. I am humbled. I have decent understanding of Newtonian physics, but when it goes to relativistic levels I am out of my depth.
Fascinating stuff. So effectively, with "infinite energy" and "unbreakable materials" we could have artificial gravity.
Artificial gravity implies you could turn it on and off. But you can't - if you have infinite energy, you also have infinite gravity, the two are bound together.
Whenever you have energy, you have gravity. I suppose you could send the energy from elsewhere, then send it back when you don't want the gravity anymore.