It is an old result of Victor Klee (answering a question of Banach) that a metrizable topological vector space (i.e., there is a translation invariant metric giving the topology) is a complete topological vector space (i.e., w.r.t. the uniformity induced by the $0$-neighborhoods) if there is some complete metric (not necessarily translation invariant) giving the same topology.

The reference is: Victor Klee, *Invariant metrics in groups (solution of a problem of Banach)*, Proc. Amer. Math. Soc. 3, 484 - 487 (1952).

The proof (which is quite similar to the arguments in Nate's answer) can also be seen in Koethe's book *Topological Vector Spaces I* §15.11.