The strong Whitney embedding theorem states that any smooth (Hausdorff and second-countable) manifold can be smoothly embedded in Euclidean space. John Nash went on to show that any Riemannian manifold could be embedded into Euclidean space in such a way that the metric of the manifold would coincide with the standard dot product. This means that the Levi-Civita connection for the manifold maps to the standard connection. More generally, for a manifold $M$ with connection $\nabla$, when does there exist an embedding into Euclidean space such that the connection is mapped to the standard connection?
The standard connection is the Levi-Civita connection of the flat metric. So if you have an embedding such that the given connection is the (projection of the) flat connection then you can also induce a metric on the embedded manifold. Hence the given metric was already a Levi-Civita connection of a Riemannian metric on your manifold. Thus the question is whether a given connection is the Levi-Civita of some Riemannian metric. That was one of your earlier questions. Does that work?