ETA: Everything below is in the paper "On the maximum scatter TSP", Arkin et al, SODA 1997.
I don't know about exact answers, but here's another approach that's a little different from Suresh's suggestion of Gonzalez clustering:
Assume for simplicity that $n$ is even. For each vertex $p$, find the median of the $n-1$ distances $d(p,q)$. Form an undirected graph in which every vertex $p$ is connected to the other vertices that are at least the median distance away. The minimum degree in this graph is at least $n/2$, so by Dirac's theorem it's possible to find a Hamiltonian cycle in this graph.
On the other hand, there are $n/2 + 1$ vertices in the disk centered at $p$ with radius $d(p,q)$, too many to form an independent set in the cycle, so any Hamiltonian cycle would have to have an edge connecting some two of these vertices, of length at most $2d(p,q)$. Therefore, the Hamiltonian cycle found by this algorithm is at worst a 2-approximation to the bottleneck max cycle.
This will work in any metric space, and gives the optimal approximation ratio among algorithms that work in any metric space. For, if you could approximate better than to within a factor of two then you could solve Hamiltonian cycle problems exactly, by reducing the input graph to the Hamiltonian cycle problem into a metric space with distance 2 for every graph edge and distance 1 for every non-edge.
Probably with some care you can massage this into an approximation algorithm for paths instead of cycles.