Short answer: Yes, but pointless.
Long answer: Not really. You would have to completely redesign the way that the OS (hell, even the boot instructions) handles commands. You would have to redo the motherboard, add a lower level of architecture to your GPU, and then reprogram the whole damned thing. Something to do with a more basic design of the fundamental registers in the CPU <citation needed> This would result in all of the system resources being used by the GPU, instead of the CPU handling the background while GPU does the in-your-face stuff.
End result: System On a Chip (SOC). Basically what the thinking is behind stuff like the Raspberry Pi (which is really freaking cool, btw).
So doable, but would have to redesign everything, and the end result would not be worth the effort.
In my humble opinion.