@JonathanLove that's a surprisingly concise and elegant argument. Thanks for taking the time to share!

Perhaps this isn't your intention, but to have this line up for the usual single-hidden-layer neural networks, shouldn't we instead want $\sigma(x;w,b) = \sigma(w\cdot x + b)$ instead of what you described in your guess (which has one of the input neurons with a constant weight of -1)?

Actually, this idea using antipodal points works for spheres as well. Thus being simply-connected is not a sufficient condition to ensure an affirmative answer either.

Elegant counter-example.This idea can be pushed a bit further as well though it was perhaps implicit: The empirical measure of a pair of antipodal points on the unit circle $\delta_p / 2+\delta_{-p} / 2$ produces an expected intrinsic distance of $\pi/2$ to any point on the circle. Thus, any two "empirical measure of antipodal points" cannot be separated by distance functionals. This shows the obstruction isn't simply due to finiteness of $\Omega$ and that the obstruction can't be avoided by requiring the underlying space to be nonzero dimensional, a manifold, connected, path-connected, etc.

Thank you for mentioning it can be done C^1 (via the Nash-Kuiper theorem I presume). However, I am mainly interested in maps with more regularity. Smooth would be ideal, but C^2 would be interesting too.

I've made some edits that hopefully help. I removed the adjective "round" since it doesn't really affect my main question (but i just meant a circle as in elementary school as opposed to a topological circle). As for the Riemannian metric structures, they should all be assumed to be induced by the ambient space. And to elaborate on the situation in higher dimension: is there a smooth Riemannian submanifold of R^4 which is a nontrivially knotted topological sphere and can be smoothly isometrically embedded into R^3?