That assumes details can be arbitrarily small, which Planck refuted.

We don't have to get anywhere near the Plank scale. If you trace the 'edges' of atoms you'll end up with a finite multiple of a meter-scale or kilometer-scale measurement. And that's the end of the fractal.

Also, at an atomic scale, the word "coast" doesn't really make sense any more...

Refuted? I thought that hypothesis was far from proven.

Not Planck, but by a few others after him. https://en.wikipedia.org/wiki/Quantum_spacetime