Marshall Hodgson lays claim to the invention of the term in the introduction to his history, The Venture of Islam, Volume 1: The Classical Age of Islam (University of Chicago Press, 1975). There are sources which suggest he coined it and used it in essays and other works starting in the 1960s, but Venture does provide a complete explanation of it, part of a lengthy discussion of cultural and geographic terminology:
On the other hand, if the analogy with 'Christendom' is held to, 'Islamdon' does not designate in itself a 'civilization', a specific culture, but only the society that carries that culture. There has been, however, a culture, centred on a lettered tradition, which has been historically distinctive of Islamdom the society, and which has been naturally shared in by both Muslims and non-Muslims who participate at all fully in the society of Islamdom. For this, I have used the adjective 'Islamicate'. I thus restrict the term 'Islam' to the religion of the Muslims, not using that term for the far more general phenomena, the society of Islamdom and its Islamicate cultural traditions.…
At any rate, it is already felt improper, among careful speakers, to refer to some local event as taking place ‘in Islam’, or to a traveller as going ‘to Islam’, as if Islam were a country. The adjective ‘Islamic’, correspondingly, must be restricted to ‘of or pertaining to’ Islam in the proper, the religious, sense, and of this it will be harder to persuade some. When I speak of ‘Islamic literature’ I am referring only to more or less ‘religious’ literature, not to secular wine songs, just as when one speaks of Christian literature one does not refer to all the literature produced in Christendom . When I speak of ‘Islamic art’ I imply some sort of distinction between the architecture of mosques on the one hand, and the miniatures illustrating a medical handbook on the other even though there is admittedly no sharp boundary between. Unfortunately, there seems to be no adjective in use for the excluded sense-‘of or pertaining to’ the society and culture of Islamdom. In the case of Western Christendom we have the convenient adjective ‘Occidental’ (or ‘Western’-though this latter term, especially, is too often misused in a vaguely extended sense). ‘Occidental’ has just the necessary traits that ‘West Christian’ would exclude. I have been driven to invent a term, ‘Islamicate’. It has a double adjectival ending on the analogy of ‘Italianate’, ‘in the Italian style’, which refers not to Italy itself directly, not to just whatever is to be called properly Italian, but to something associated typically with Italian style and with the Italian manner. One speaks of ‘Italianate’ architecture even in England or Turkey. Rather similarly (though I shift the relation a bit), ‘Islamicate’ would refer not directly to the religion, Islam, itself, but to the social and cultural complex historically associated with Islam and the Muslims, both among Muslims themselves and even when found among non-Muslims.
The pattern of such a double adjectival ending, setting the reference at two removes from the point referred to, is sufficiently uncommon to make me hesitate. But there seems no alternative. In some contexts, but only in some, one can refer without ambiguity to the 'Perso-Arabic' tradition to indicate 'of or pertaining to' Islamdom and its culture, for all the lettered traditions of Islamdom have been grounded in the Arabic or th Persian or both. In other cases, one might use a periphrasis involving the terms 'traditions/culture/society of Islamdom'. One cannot, speaking generally, call Swedish 'a Christian language'; and if one were debarred from calling it an 'Occidental' language, one could not say simply that it is 'a language of Christendom', which might in some contexts seem to imply that it was to at least some extent used throughout that extensive realm; but one might say it is 'a language of the culture of Christendom'.
This level of distinction is not commonly needed or even comprehended outside of area studies, and the term is quite rare in non-academic publications. It does not turn up at all in several of the BYU corpora including Google Ngrams, BNC, or Hansard, and its only entry in COCA is from an art journal from 2009. The News on the Web Corpus turns only 22 entries, all since 2010— a new edition of The Venture of Islam was published in 2009. By comparison, ummah variously spelled has not quite 300 results, and Islam has 12,304.
The website for the 2014 Islamicate Studies Symposium, held on the occasion of the 40th anniversary of Hodgson's work, notes that he was the originator of "a number of neologisms and some very careful usages," Islamicate being one of them.