"Eastern Hemisphere" and "Western Hemisphere" are certainly well established terms in English, though a quick check in my English-French and English-German dictionaries suggests that other language communities probably only recognize Northern and Southern.
When we talk about the Northern and Southern Hemispheres, we can define them exactly by reference to the Equator. Also, there are easily observable basic differences between North and South. It's summer at present in Australia but it's winter here in Europe. I've also been told that bath-water going down the plug-hole will swirl clockwise in one hemisphere and counter-clockwise in the other, but I can't remember which is which.
But Eastern and Western Hemispheres are much vaguer terms. The OED and Chambers agree that the Eastern Hemisphere is the "half" that contains Europe, Africa and Asia and the Western is the "half" that contains the Americas.
These definitions, and the quotations in the OED, suggest that the terms came into use in the wake of the rediscovery of America by Europeans, and that they mean little more than "the East" and "the West", with the dividing line being very vague. The Americas were, as far as people knew, "half a world away" from Europe, and it was natural for Europeans to think of the world as being divided longitudinally somewhere off the Atlantic coast of Europe and Africa (the division obviously doesn't coincide with the meridian of Greenwich, otherwise I'd be moving between hemispheres when I travelled three stops westwards from home on the London Underground). Is Australia in the Eastern or Western hemisphere? I don't know, probably because Australia was unknown to the persons who first used the terms.
In a sermon in 1624, John Donne spoke of "The Western Hemisphere the land of Gold and Treasure; The Eastern Hemisphere the Land of Spices and Perfumes".
The differences between the "Old World" and "New World" were probably big enough to justify thinking of them as separate hemispheres when the Americas were rediscovered, and we may still think that there are considerable cultural differences.
But the differences between the vaguely defined Eastern and Western Hemispheres are not of the same order as those between the Northern and Southern, and so the French may be justified in refusing to believe in "Eastern" and "Western" Hemispheres. Indeed, if they can read the magnificent seventeenth-century English of Sir Thomas Browne, they can find the arguments against such an idea in Chapter 7 of Book 6 of Browne's "Pseudodoxia Epidemica". I'm eternally indebted to Bruyere for raising this topic, because it's led me to find a complete text of Browne's work on the web at http://penelope.uchicago.edu/