Is Hamiltonian path NP-hard on graphs of diameter 2?
Assuming you are talking about directed graphs, the work of Füredi et al. [1] shows that such graphs of n vertices and diameter 2 have at least ~n log (n) edges, or an average degree of ~log(n).  I am aware of no subsequent result tightening this result, and the authors themselves suggest that they think this result is tight; this is quite far from the known sufficient conditions for Hamiltonian paths, which are based on a roughly linear vertex degree.

As a result, this appears to be an open and difficult problem that would require some significant research for a solution.

[1] Füredi, Zoltán, et al. "Minimal oriented graphs of diameter 2." Graphs and Combinatorics 14.4 (1998): 345-350.