Demazure and Gabriel's Introduction to algebraic geometry and algebraic groups develops standard algebraic geometry in terms of functors on CRing.

Martin Olsson's stacks course did something like what you're describing, first characterizing separated schemes among functors on Aff, then algebraic spaces among functors on Sch, then algebraic stacks among stacks over Sch. I took notes in that class which I think are pretty good: http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf

Demazure and Gabriel's Introduction to algebraic geometry and algebraic groups develops standard algebraic geometry in terms of functors on CRing.

Martin Olsson's stacks course did something like what you're describing, first characterizing separated schemes among functors on Aff, then algebraic spaces among functors on Sch, then algebraic stacks among stacks over Sch. I took notes in that class which I think are pretty good: http://stacky.net/files/written/Stacks/Stacks.pdf