This is not a question about MATLAB, but since it already has an answer...
Yes. You should be able to prove that the maximal diameter must lie between two vertices of the polygonal region. I would do that by starting with a pair of parallel lines in the plane that do not intersect the polygon, but contain the polygon between them. You can move them closer together, until they just touch the polygonal region. Clearly that event will happen when one or more of the vertices of the polygon lie along each of the parallel enclosing lines. The diameter of the polygon in that direction is just the distance between the lines.
Now rotate the lines you would consider, implicitly creating a function of the dameter of the polygon at any rotation angle. Clearly this function is continuous, though not differentiable. Still it will have a well defined maximum. That maximum need not be unique (consider a square).
In the end, you will have shown that the maximum diameter will be given by the maximum Euclidean distance between any pair of vertices of the polygon, so pdist, applied to the polygonal vertices, will suffice.