A consistent path system in a graph $G$ is an collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We say that $G$ is strictly metrizable if every consistent path system in $G$ can be realized as the system of unique geodesics with respect to some assignment of positive edge weight. In this paper, we show that the family of strictly metrizable graphs is minor-closed.