How true is this?

This seems to be an urban myth, although a strangely pervasive one. On chronological grounds alone, coffee wasn't a widely-consumed drink in the Ottoman Empire until the sixteenth century—Ralph S. Hattox, in his classic study of coffee and coffeehouses in the empire, estimates that it reached Istanbul "around the middle of the 1500s." Note that the same authors who make the divorce claim also tend to claim that the first coffeehouse in Istanbul, which they call Kiva Han, opened in the mid-to-late fifteenth century (1475 is a popular date). But since I haven't seen any reliable citations for this factoid and Hattox wrote his dissertation on Mamluk and Ottoman coffeehouses, so I'm inclined to trust him on this one. (The İslam Ansiklopidesi backs up Hattox on this point: "it is generally accepted that the first coffeehouse in Istanbul opened in the early 1550s.")

It's also difficult to judge exactly how much of the coffee consumed in early modern Istanbul was drunk in the home. While Cemal Kafadar notes that coffee in time did become part of the domestic breakfast ritual, he also argues that coffeehouses (for men—the female equivalent were the public baths) satisfied the majority of Turkish demand for coffee. In fact, the earliest evidence Kafadar mentions for the domestic consumption of coffee (albeit at a level that indicates it was "becoming a regular part of everyday life in homes" by that point) is for the seventeenth century!

At the risk of bringing the "common sense" banhammer upon myself, I would suggest that the absorption of demand by social institutions, the anachronism of the 15th-century date, the fact that no academic sources seem to cite this alleged element of the Ottoman divorce law, and the strained relationship early modern Ottoman authorities had with the beverage (which probably has more to do with the perceived immorality and unruliness of the coffeehouses and less with attitudes towards the drink itself)... all these would mitigate against me accepting the divorce claim as even remotely true.

Sources:

Ralph S. Hattox, Coffee and Coffeehouses: The Origins of a Social Beverage in the Medieval Near East

Cemal Kafadar, "A History of Coffee," in The XIIIth Congress of the International Economic History Association (IEHA), 22-26.

Cemal Kafadar, "How Dark is the History of the Night, How Black the Story of Coffee, How Bitter the Tale of Love: The Changing Measure of Leisure and Pleasure in Early Modern Istanbul," in Medieval and Early Modern Performance in the Eastern Mediterranean, ed. Arzu Öztürkmen and Evelyn Birge Vitz

İdris Bostan, "Kahve," in İslam Ansiklopidesi

İdris Bostan, "Kahvehane," in İslam Ansiklopidesi