Given a sufficiently smooth manifold $M$,
a Riemannian metric on $M$ induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely)
an embedding of $M$ into Euclidean space induces a Riemannian metric on $M$ by pullback of the induced metric. (naturally)
So, there is at least a surjection from the set of Euclidean embeddings (say, for fixed target dimension) to the set of Riemannian metrics on $M$, in the appropriate smoothness categories.
Is there more structure to this correspondence? For instance, could it be continuous in some cases, perhaps with the appropriate topologies and after appropriate quotients on the space of metrics and the space of embeddings?
The previous question Smooth isometric embeddings of Riemannian manifolds notes, among other things, that the isometric embedding can be quite rigid in some cases (ex: positive curvature metrics on $S^2$ embed uniquely in $\mathbb R^3$, presumably modulo Euclidean motions).
EDIT(0): corrected per @MichaelAlbanese's initial comment.
EDIT(1): cross-posted: https://mathoverflow.net/questions/374097/correspondence-between-riemannian-metrics-and-euclidean-embeddings