Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset? What about the case in which the ambient manifold is an euclidean space?

It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).

Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $\mathbb{R}^3$. In particular, this immersion is not proper and so the image is not a closed subset of $\mathbb{R}^3$. In contrast, Colding-Minicozzi showed that any complete embedded minimal surface in $\mathbb{R}^3$ (i.e. a two dimensional submanifold of $\mathbb{R}^3$ in the usual sense) that has finite topology must be properly embedded and hence be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of difficult open problems about exactly how general this phenomena is -- e.g. is it true for surfaces of finite genus? Any complete embedded minimal surface?