You may want to have a look at the article Foliation by constant mean curvature spheres, by Rugang Ye, Pacific Journal of Mathematics, 147 (1991), 381–396.

In this article, the author shows, given a Riemannian surface $M$, that every nondegenerate critical point $p$ of the Gauss curvature has a punctured neighborhood that is foliated by closed curves of constant geodesic curvature. (Think of the rings of a bulls-eye centered on $p$.)

In particular, for an ellipsoid with three distinct axes, the six critical points of the Gauss curvature (where the axes meet the surface) are nondegenerate, and hence there exists a foliation by closed, constant geodesic curvature curves centered on each one. These must be real-analytic, so I imagine that the closed constant geodesic curvature foliations around each axis extend to foliate the entire ellipse away from the corresponding axis points. This would give three distinct $1$-parameter families of closed constant geodesic curvature curves on the ellipsoid with three distinct axis lengths. I don't know whether the classic differential geometers were able to compute these curves explicitly or not.

Of course, there will be many other closed curves with constant geodesic curvature on the ellipsoid that don't fall into these families. (Just think of the ones with geodesic curvature $0$, i.e., the closed geodesics, for example.) However, the 'generic' curve of constant geodesic curvature on the ellipsoid does not close.

On the other hand, it was asserted (without explicit proof) by Darboux (see Livre VI, Chapitre VII, Paragraph 622, footnote 1 of his Leçons sur la Théorie Générale des Surfaces) that the only surfaces for which all of the curves of constant geodesic curvature are closed are the surfaces of constant Gauss curvature. I don't know where a proof might first have appeared in the literature, but I once asked Victor Guillemin about this, and he produced a proof. In Ye's article quoted above, he shows that, if one has a sequence of closed curves of constant geodesic curvature whose curvatures go to infinity and that converge to a point $p$ in the surface, then $p$ must be a critical point of the Gauss curvature, and this provides an alternative proof of Darboux' claim.

You may want to have a look at the article Foliation by constant mean curvature spheres, by Rugang Ye, Pacific Journal of Mathematics, 147 (1991), 381–396.

In this article, the author shows, given a Riemannian surface $M$, that every nondegenerate critical point $p$ of the Gauss curvature has a punctured neighborhood that is foliated by closed curves of constant geodesic curvature. (Think of the rings of a bulls-eye centered on $p$.)

In particular, for an ellipsoid with three distinct axes, the six critical points of the Gauss curvature (where the axes meet the surface) are nondegenerate, and hence there exists a foliation by closed, constant geodesic curvature curves centered on each one. This gives three distinct $1$-parameter families of closed constant geodesic curvature curves on the ellipsoid with three distinct axis lengths. I don't know whether the classic differential geometers were able to compute these curves explicitly or not.

Of course, there will be many other closed curves with constant geodesic curvature on the ellipsoid that don't fall into these families. (Just think of the ones with geodesic curvature $0$, i.e., the closed geodesics, for example.) However, the 'generic' curve of constant geodesic curvature on the ellipsoid does not close.

On the other hand, it was asserted (without explicit proof) by Darboux (see Livre VI, Chapitre VII, Paragraph 622, footnote 1 of his Leçons sur la Théorie Générale des Surfaces) that the only surfaces for which all of the curves of constant geodesic curvature are closed are the surfaces of constant Gauss curvature. I don't know where a proof might first have appeared in the literature, but I once asked Victor Guillemin about this, and he produced a proof. In Ye's article quoted above, he shows that, if one has a sequence of closed curves of constant geodesic curvature whose curvatures go to infinity and that converge to a point $p$ in the surface, then $p$ must be a critical point of the Gauss curvature, and this provides an alternative proof of Darboux' claim.