No, the Einstein field equations actually wrap things up pretty neatly as far as why it happens (although not how). Energy, momentum, stress, and mass are all the same. Energy turns into matter and vice versa. E=mc^2.

Going faster doesn't change your velocity relative to light, just your reference frame. In the same way, getting heavier changes your reference frame. If you're changing your relative time, you're also changing how far you're traveling from an external reference. And if the distance is changing, it means space is warping... which is exactly what gravity is. Energy/mass/momentum warps space and that gives us gravity.