The property in the title is wellknown. I am trying to find an original reference to its first appearance in print. The 4th edition of Graphs & Digraphs by Chartrand and Lesniak lists this as Theorem 13.6 and says that it's a generalization of a result by Gilbert, but gives no further reference.
I think you will find it in Moon, J. W.; Moser, L. Almost all (0,1) matrices are primitive. Studia Sci. Math. Hungar. 1 (1966) 153–156. But I don't have time to visit the library to be sure and I don't see it online.
It is certainly in Burtin, Ju. D. Asymptotic estimates of the diameter and the independence and domination numbers of a random graph. (Russian) Dokl. Akad. Nauk SSSR 209 (1973), 765–768.
I guess the Gilbert mentioned is Gilbert, E. N. Random graphs. Ann. Math. Statist. 30 (1959) 1141–1144. It isn't clear exactly why...

$\begingroup$ The introductory paragraph of a 1981 TAMS paper by Bollobas available at ams.org/journals/tran/198126701/S00029947198106219717/… supports the likely MoonMoser origin. Also, an author search on "e.n. gilbert" at projecteuclid.org will produce his "Random Graphs" article. $\endgroup$ Dec 3 '12 at 17:57

$\begingroup$ I checked the Moon Moser paper, that is indeed your reference. I've emailed you a somewhat crappy but legible scan of the paper. $\endgroup$ Dec 3 '12 at 19:59
The result you asked about follows instantly from Fagin's proof of the zeroone law for finite graphs. He shows that all of Gaifman's extension axioms have asymptotic probability 1, and "diameter $\leq 2$" is essentially one of the extension axioms. Fagin's paper is "Probabilities on finite models" [J. Symbolic Logic 41 (1976) pp.5058]. I believe the zeroone law was proved earlier by four Russians, but I don't have access to their paper and don't know whether their method immediately implies the "diameter $\leq2$" result.

$\begingroup$ The four Russians are Y.V. Glebskii, D.I. Kogan, M.I. Liogon'kii, and V.A. Talanov. The paper is "Range and degree of realizability of formulas in the restricted predicate calculus" [Kibernetika (Kiev) 1969, no.2, 1728; translation in Cybenetics (Kiev) 5 (1969) 142154]. I've been told that this paper is rather difficult to read. $\endgroup$ Dec 3 '12 at 13:58

$\begingroup$ Thanks! I actually know this derivation, but I wanted a encapsulated reference to this precise fact. $\endgroup$ Dec 3 '12 at 14:03