This strikes me as completely wrong. Who is to say there is an upper limit on how big the computer running the universe can be, or that it has to be Turing complete?

And why on Earth would that imply the laws of physics would work differently at different scales. "Scale" is just an abstraction we use to make sense of things, not a fundamental property of the universe (so far as I know.) If you zoom out of Conway's game of life enough, everything looks completely different as if it was operating by a different set of rules.