Did George Washington predict a "United States of Europe"?

No.

This spurious quotation attributed to George Washington has been promoted recently by a poster at Majority Rights: “I am a citizen of the greatest Republic of Mankind. I see the human race united like a huge family by brotherly ties. We have made a sowing of liberty which will, little by little, spring up across the whole world. One day, on the model of the United States of America, a United States of Europe will come into being. The United States will legislate for all its nationalities.” Variations have also been repeated by Belgian Prime Minister Guy Verhofstadt, the half-Japanese founder of the Pan-European Union Coudenhove-Kalergi, Eric Voegelin, and others.

For the search engines, I'm reposting my reply from Majority Rights showing the source of this bogus Washington quotation below (continue reading):

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."2 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 constitided the United States of Europe."1

Title The people of action: an essay on American idealism
Authors Gustave Rodrigues, James Mark Baldwin
Translated by Louise Seymour Houghton
Publisher C. Scribner's Sons, 1918
http://books.google.com/books?id=b8Y9AAAAYAAJ

This is an English translation of an essay by "Frenchman" Gustave Rodrigues. Rodrigues, in turn, cites a biography of Washington by another Frenchman. Here is the transcription (from the Library of Congress website, which also hosts images of the letter) of the actual letter from Washington to Lafayette that is cited:

Altho' I pretend to no peculiar information respecting commercial affairs, nor any foresight into the scenes of futurity; yet as the member of an infant empire, as a Philanthropist by character, and (if I may be allowed the expression) as a Citizen of the great republic of humanity at large; I cannot help turning my attention sometimes to this subject. I would be understood to mean, I cannot avoid reflecting with pleasure on the probable influence that commerce may hereafter have on human manners and society in general. On these occasions I consider how mankind may be connected like one great family in fraternal ties. I indulge a fond, perhaps an enthusiastic idea, that as the world is evidently much less barbarous than it has been, its melioration must still be progressive; that nations are becoming more humanized in their policy, that the subjects of ambition and causes for hostility are daily diminishing, and, in fine, that the period is not very remote, when the benefits of a liberal and free commerce will, pretty generally, succeed to the devastations and horrors of war.

Some of the late treaties which have been entered into, and particularly that between the King of Prussia and the Ud. States, seem to constitute a new era in negotiation, and to promise the happy consequences I have just now been mentioning. But let me ask you my Dr. Marquis, in such an enlightened, in such a liberal age, how is it possible the great maritime powers of Europe should submit to pay an annual tribute to the little piratical States of Barbary? Would to Heaven we had a navy able to reform those enemies to mankind, or crush them into non-existence.

I forbear to enter into a discussion of our domestic Politics, because there is little interesting to be said upon them, and perhaps it is best to be silent, since I could not disguise or palliate where I might think them erroneous. The British still hold the frontier Posts, and are determined to do so. The Indians commit some trifling ravages, but there is nothing like a general or even open war. You will have heard what a loss we have met with by the death of poor Genl. Greene. General McDougal and Colo. Tilghman are also dead.

This ["citizen of humanity" talk] is typical enlightenment pablum [and nothing more]. No talk of a "United States of Europe" here, nor "elsewhere" from Washington. The other quotation Rodrigues attributes to Washington is merely the 19th-century French biographer putting words into the mouths of "Washington and his friends" without citing any source, Washington having never written anything like this:

Les États-Unis garantissent à chaque État admis
dans l'Union une forme républicaine de gouvernement ;
ils le protègent contre l'invasion ; ils le défendent, .«ar
la denaande de ses représentants, contre toute violence
domestique; ils le rendent participant des avantages
de la société commune ; et ils font jouir tous les
citoyens des droits essentiels de la personne humaine.

Washington et ses amis disaient :

« Notre exemple prouvera aux hommes qu'ils ne
sont pas condamnés à recevoir éternellement leur
gouvernement du hasard et de la force, et qu'ils sont
capables de se donner de bonnes institutions par
réflexion et par choix.

» 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 États-Unis d'Amérique,
se constitueront les États-Unis d'Europe. »

La constitution votée par la Convention américaine
commença à être appliquée en 1789.

La triple élection des députés, des sénateurs et du
présidentse fit pacifiquement. A l'unanimité, Washing-
ton fut nommé président des Etats-Unis.

Washington, libérateur de l'Amérique: suivi de Washington et la révolution américaine ... (1886)
Author: Joseph Fabre
Publisher: C. Delagrave
Year: 1886
http://www.archive.org/stream/washingtonlibra00fabrgoog/washingtonlibra00fabrgoog_djvu.txt