This is indeed generally true. The proofs I know use geometry in one way or another.
Let me explain the case of a leaf $L$ of $\mathcal F$ that has no singularities.
Let's fix a hyperbolic structure on $M$, and so the universal cover $\tilde M$ is isometric to the hyperbolic plane $\mathbb H^2$. If you understand the connection between measured foliations and geodesic laminations then you will know that corresponding to $\mathcal F$ there is a geodesic lamination $\mathcal L$ on $M$, there is a self-homotopy $h : M \times [0,1] \to M$, and there is a leaf $L'$ of $\mathcal L$, such that $h \mid L' \times [0,1]$ is a homotopy from the identity map to $L$. It follows that in the universal covering space there are lifted leaves $\tilde L$ and $\tilde L'$ are properly homotopic. Since $\tilde L'$ is a bi-infinite geodesic in $\mathbb H^2$, it is a closed subset, hence $\tilde L$ is a closed subset.
Another proof uses geometry in a different way. Using that $\mathcal F$ is minimal, one can prove that there exists another measured foliation $\mathcal F'$ which is transverse to $\mathcal F$. The two transverse measure of $\mathcal F$ and $\mathcal F'$, taken together, determine a conformal structure $\mu$ on $M$ and a quadratic differential $q$ of $\mu$. There is an induced singular Euclidean metric on $M$: in regular local coordinates where $q = dz^2$ the metric is given by the formula $dx^2 + dy^2$ where $z=x+iy$; in singular local coordinates where $q = z^k dz^2$ ($k \ge 1$), the formula is a bit more complicated. The leaf $L$ is a bi-infinite geodesic in this singular Euclidean structure. The lift to $\tilde M$ of the singular Euclidean structure is a complete $\text{CAT}(0)$ geodesic metric, and $\tilde L$ is a complete bi-infinite geodesic in this structure, hence $\tilde L$ is a closed subset.
