The word “Asia” comes from Greek, as does Europe, both of “unknown origins”.
The word “America” was “apparently first used in M. Waldseemüller Cosmographiae Introductio (1507) < Americus, Latinized form of the name of Amerigo Vespucci (1451–1512), Italian explorer who navigated the coast of South America in 1501.”
Australia is from Latin, “austr{amac}lis, in Terra Australis ‘southern land,’ the title given, from 16th c., to the supposed continent and islands lying in the Great Southern Ocean, for which Australia was at length substituted (see Flinders, 1814, Voyage to Terra Australis, I. Introd. p. iii, foot-note.)”
Antarctica is from Old French:
“OFr. antartique (= Pr. antartic, It. antartico), ad. L. antartic-us, – arctic-us, a. Gr. {alenis} {nu} {tau} {alpha} {rho} {kappa} {tau} {iota} {kappa} – {goacu} {fsigma} opposite to the north, f. {alenis}{nu}{tau}{giacu} against, opposite + {alenis}{rho}{kappa}{tau}{iota}{kappa}-{goacu}{fsigma} of the Bear, northern, f. {alenisacu}{rho}{kappa}{tau}{omicron}{fsigma} bear, the constellation of the Bear. The orig. Eng., phonetically modified by passage through Romance, has, like mod.Fr. antarctique, been since conformed to the Gr. spelling, though still often pronounced (æn{sm}t{fata}{lm}t{shti}k).”
Africa is from Old English
“OE. (only pl.) Africanas, ad. L. Afric{amac}nae, f. Africa, n. use of fem. (sc. terra land) of Africus, f. Afr{imac} (sg. Afer) ancient people of N. Africa;”
My source is the Oxford English Dictionary. I apologize that the International Phonetic Aphabet symbols did not copy/paste very well. Those would be the seemingly random assortment of characters inside of {}.
In my opinion, the OED is a close to the origins of these words as one will find outside of becoming a linguistics PhD, and doing his or her own research on their origins.