Apologize is the preferred spelling in American and Canadian English, and apologise is preferred in varieties of English from outside North America. This is the case despite the fact that apologize is the original form and was once standard even in British English (and is still used by some British publishers).
Traditionally, verbs whose roots have origins in Greek take the -ize suffix, and apology is of Greek origin, so it took -ize when it first came to English in the 16th century. Apologise emerged as a variant fairly early in the word’s history, but it only recently became the preferred spelling outside North America, possibly due to a mistaken belief that apologize is the American spelling and hence to be avoided. A similar trend has guided the recent evolution of the traditional realize into the newer realise in varieties of English from outside North America.
Apologize is preferred in North American publications—for example:
Nearly a third spontaneously apologized for their crimes and showed signs of true repentance. [Toronto Star]
The fine is to be even higher if Chevron doesn’t publicly apologize. [Wall Street Journal]
Executives from a Japanese Sony music unit apologized Wednesday for a rock band under its management that dressed up like Nazis on a national TV broadcast. [Winnipeg Free Press]
British discount carrier easyJet is apologizing to Jewish customers after it stocked only bacon baguettes and ham melts as meal selections on a flight from Tel Aviv earlier this month. [USA Today]
Australian and British publications prefer apologise—for example:
I am not ashamed to say that I was too flabbergasted to apologise. [Telegraph]
It is not the first time Ms Ferguson has had to apologise for her financial dealings. [Herald Sun]
Clare this week apologised to Jessica for any hurt. [Guardian]