I have been playing around with space-filling curves that completely fill the unit square. All of them that I have seen have a fractal dimensional of 2. Makes sense that it would be 2, but a Google search hasn't turned up any comments on this at all. So, is this a general result? Thanks.
You seem unclear about the definitions. A curve that fills the unit square is a continuous map $\gamma : [0,1] \rightarrow [0,1]^2$ such that $\gamma([0,1])=[0,1]^2$. For a general curve (not necessarily filling the square), the Hausdorff dimension of that curve is simply the Hausdorff dimension of $\gamma([0,1])$, i.e. of the image of the curve. Since the Hausdorff dimension of the unit square is 2, then yes, by definition the dimension of a curve filling the unit square is 2.