“Just consider, all the little nations running for shelter here and there, one running to Russia and another to the United States. In that situation before anything else would happen, the world will have been polarised between the Soviet Union and the United States. It is against that negative polarisation we have been fighting for years. We want to have the opportunity to interpose between these two giants a moderating, modifying and mitigating diplomacy.”

On the Irrepressible Conflict (1858)
Context: The Union is a confederation of States. But in another aspect the United States constitute only one nation. Increase of population, which is filling the States out to their very borders, together with a new and extended network of railroads and other avenues, and an internal commerce which daily becomes more intimate, is rapidly bringing the States into a higher and more perfect social unity or consolidation. Thus, these antagonistic systems are continually coming into closer contact, and collision results.
Shall I tell you what this collision means? They who think that it is accidental, unnecessary, the work of interested or fanatical agitators, and therefore ephemeral, mistake the case altogether. It is an irrepressible conflict between opposing and enduring forces, and it means that the United States must and will, sooner or later, become either entirely a slaveholding nation, or entirely a free-labor nation.

2019

1970s, Remarks on Being Reelected (1972)

Attributed to Washington in "Farewell to the United States of Europe: long live the EU!" by André Fontaine at Open Democracy (29 November 2001) http://www.opendemocracy.net/democracy-europefuture/article_344.jsp. It appears to have originally circulated in French:
:: Je suis citoyen de la Grande République de l'Humanité. Je vois le genre humain uni comme une grande famille par des liens fraternels. Nous avons jeté une semence de liberté et d'union qui germera peu à peu dans toute la Terre. Un jour, sur le modèle des Etats-Unis d'Amérique, se constitueront les États-Unis d'Europe. Les États-Unis seront le législateur de toutes les nationalités.
: An anonymous blogger in "Did George Washington predict a "United States of Europe"? (30 January 2010) http://racehist.blogspot.com/2010/01/did-george-washington-predict-united.html showed that it derived from Gustave Rodrigues, Le peuple de l'action: essai sur l'idéalisme américain (A. Colin, 1917), p. 207:
:: Washington écrivait à La Fayette qu'il se condérait comme « citoyen de la grande république de l'humanité » et ajoutait : « Je vois le genre humain uni comme une grande famille par des liens fraternels ». Ailleurs il écrivait, prophétiquement: « Nous avons jeté une semence de liberté et d'union qui germera peu à peu dans toute la terre. Un jour, sur le modèle des Etats-Unis d'Amérique, se constitueront les États-Unis d'Europe. »
: A translation by Louise Seymour Houghton ( The People of Action: An Essay on American Idealism (1918) http://books.google.com/books?id=b8Y9AAAAYAAJ) reads:
:: Washington wrote to Lafayette that he considered himself a "citizen of the great republic of humanity," adding: "I see the human race a great family, united by fraternal bonds." Elsewhere he wrote prophetically: "We have sown a seed of liberty and union that will gradually germinate throughout the earth. Some day, on the model of the United States of America, will be constituted the United States of Europe." [pp. 209-210]
: The first two quotations come from a letter to the Marquis de Lafayette of 15 August 1786 (see above) as quoted in Joseph Fabre's Washington, libérateur de l'Amérique: suivi de Washington et la revolution Américaine (Ch. Delagrave, 1886), and the third is also found in that source where, although placed between quotation marks, it is clearly intended as the author's own comments on what "Washington and his friends" were saying to the world by establishing the American Constitution. Gustave Rodrigues mistakenly printed Fabre's words as Washington's alongside some actual observations of his from a letter to Lafayette, and so created the misquotation.
Misattributed

Progress, Coexistence and Intellectual Freedom (1968), Dangers, The Threat of Nuclear War

Barack Obama: "The President's News Conference With President Abdullah Gul of Turkey in Ankara, Turkey," April 6, 2009. Online by Gerhard Peters and John T. Woolley, The American Presidency Project. http://www.presidency.ucsb.edu/ws/index.php?pid=85974&st=&st1=
2009

'
Source: Made in Japan (1986), p. 173.

and with Britain in 1948 and 1956
Progress, Coexistence and Intellectual Freedom (1968), Dangers, Vietnam and the Middle East

Remarks at a meeting with Serb leaders (16 March 1991), as quoted in Doder and Branson (1999) Milosevic: Portrait of a Tyrant