The following is a history of both the word acclimate and acclimatise (or acclimatize), as listed on the Online Etymology Dictionary:
Acclimate: 1792, from French acclimater, verb formed from à "to" + climat. Related: Acclimated; acclimating.
Acclimatise: 1836; a more recent formation than acclimate and replacing it in many uses.As we can see, acclimate—the predominant usage in American English—predates that of acclimatise by some 44 years.
Notice I used the word "predominant." This is because neither usage is 100% exclusive to Britain or the United States and, indeed, each has enjoyed a level of popularity on either side of the Pond since their respective introductions to the language. Actually, within American English, both acclimatize (perceived U.S. spelling. More on that later.) and acclimate had spent the better part of 150 years neck and neck in the battle for supremacy. It was not until the 1970s that acclimate finally began to pull away from its long-time rival to earn predominant status (see the Ngram below).
Ngram: American English
Intriguingly, the race for predominance has not been anywhere near as close in British English. Since the 1850s, the word acclimatise has stormed ahead of acclimate, with peak usage of the former coming in the early twentieth century. That said, there has been something of a resurgence of acclimate in British English since around the mid-1960s (see Ngram below), possibly due to an influx of American popular culture.
Ngram: British English
Meanwhile, when we compare acclimatise and acclimatize (the perceived U.S. spelling), we make an interesting discovery. While it is true that Americans almost always incorporate -ize, the same cannot be said of the British when it comes to -ise.
Remember when I said that acclimatise peaked in the early twentieth century? Well, that's because at around this time, acclimatize (with a "z") was starting to take flight in British English. Indeed not only did it eventually overtake the alternative spelling—peaking shortly after World War II—but it remains the predominant spelling in British English to this day. See the final Ngram below.
Ngram: British English
Which do YOU say? Acclimatise or acclimate? Let us know in the comments below.