I think I would make a case for neither. The most plausible time period in which you could speak of an "Islamic Empire" is from the expansionary period after the death of Muhammad until the splitting of that empire as a result of the Abbasid revolution (i.e. 632-750). But the extent to which you could call that empire "Islamic," particularly in the first half of that time period, is debatable. And to call the territory they ruled an Oslamic civilization seems untenable given that it was overwhelmingly non-Muslim for at least two centuries after the rise of Islam, if not longer.

In general and certainly for the late antique and medieval periods, I prefer Marshall Hodgson's half-century old term "Islamdom" which correlates to the Christian equivalent of Christendom.

I'm terms of "civilizations", as in Huntington's famous "clash of civilizations", I'm not a theorist but I'm somewhat unconvinced of the usefulness of the term. If nothing else it seems to me that that there is as strong a claim to there being an "Arab civilization" or "Persian civilization" as there is to an overarching "Islamic civilization." As a result I quite often prefer "The Islamic world" or "Arabic-speaking part of the Islamic world" or "Arab world" and so-forth, although some of this will be a matter of preference.