Let $G$ be a Lie group with a **left** invariant metric $g$.

Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is **right**-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $h \in H$). Note that this is equivalent to the statement, that for every $h \in H$ the map of right multiplication by $h$, viewed as a map $G \mapsto G$ is an isometry.

(This appears to be stronger thant requiring it will be an isometry as a map $H \mapsto H$).

Is it true that $H$ must be totally geodesic in $G$?

Since $g$ is bi-$H$-invariant, the geodesics of $H$ are the one-parameter subgroups of $H$. Hence, the question amounts to:

Are the one-parameter subgroups of $H$ geodesics in $G$?