...but I'm not comfortable with the statement that it cannot be proved by measuring physical triangles...

It depends on what you mean by "proved". The Pythagorean Theorem applies to all possible triangles (on a flat Euclidean plane), and the answer it gives you is infinitely precise. If you are measuring real triangles on Earth, however, the best you could do is get close to the answer, due to the uncertainty inherent in your instruments (among other factors). Still, you could very easily disprove a theorem that way, and you could also use your experimental results to zero in on the analytical solution much faster than if you were operating from pure reason alone.