Allah does not equate to "God", . . .
Looks like it's time for some basic Arabic lessons around here.
If one is speaking Arabic, Allah -- an elision of Al Illah, meaning literally "the god" -- is what any monotheist, Christian, Muslim or Jew, calls the one existing deity. The definite article elided with the general word for a god has the same force in Arabic as capitalizing god to yield God does in English. In Greek the same applies 'o theos, "the god" means the same as the English God. For instance the first phrase of the Orthodox Christian Trisagion prayers, in English "Holy God, Holy Mighty, Holy Immortal have mercy on us!" in Greek " 'Agios 'o Theos, 'Agios ischyros, 'Agios athanatos, eleison imas!", runs in Arabic "Quddūsun Allāh, Quddūsun al-qawī, Quddūsun alladhī lā yamūt urḥamnā."
It is perfectly sound to translate "Allahu Akbar" from Arabic to English as "God is greater".
However, only Muslims will refer to God (as misconceived in their theology) as "Allah" when speaking languages other than Arabic (or Malay, where the Arabic word was adopted), following Mohammed's vain conceit that the Arabic word for God is the proper name of the one existing deity. A Christian who is bilingual Arabic-English will say "Allah" in Arabic and "God" in English, a bilingual Muslim will say "Allah" in both.
Of course McCain is being a complete fool here -- the phrase "Allahu Akbar" is a war cry, its first use being by Mohammed's troops at the Battle of Badr in 624. The nearest parallel I can think of for the phrase "Thank God" would be the line in the Agincourt Carol, "Deo gratias Anglia redde pro victoria!"