I have looked upon many common English surnames from the census records of late 1800s and 1900s, and found out that they are either most likely of English (Old English, Middle English), French (Norman), Scottish, Irish, Norse, or German (Old German) in origin. Some common English surnames that i have encounter are Dutch (Flanders), and Scandinavian, but these are much rarer in England. The connection is that the origins of most surnames in England are not that far out, but close neighboring countries. To have a common English surname from the late 1800s whos origin is from the other side of Europe is extremely unlikely (I don't think there is any). With that said, I have doubts that the English surname Gazey, quite a common surname back in 1800s, has its origins all the way back in Romania, Bulgaria, and Yugoslavia.
I believe this surname Gazey is of English origin, possibly originating from France, quite a common surname in late 1800s in county Warwickshire, England. There are no Gazey families found in Southeastern Europe, but I could see why that origin has been suggested.
The English Gazey is pronounced "gAYzee" as in English surnames Gaze and Gazeley, but the southeastern "Gazey" I think is pronounced "gAHzee" as in Gozz. This "Gazey" surname from southeastern Europe I think is ultimately of Arabic origin, spelt as Gazi, Ghazi, definitely originated from "Ghaz" but the second part "ey" of the name is uncertain, most likely derived from "i" in the name Ghazi. (sounds almost like Gazey). This surname Ghazi is the closest surname I could find that is similar to the name Gazey in spelling. The origins of the name Ghazi is found here - (http://www.ancestry.com/name-origin?surname=Ghazi) Another similar surname is Arabic Ghazal, origins are found here - (http://www.ancestry.com/name-origin?surname=Ghazal)
Now, the Bulgarian Gazey, a name for a mountain and lakes, perhaps is of different origin. It is derived from Bulgarian Газей, which in google translate, translates into "gazette". The origins behind the name Gazey for the mountain and how long it had the name I have not found yet, but I don't think there are any Gazey bearers that are from Bulgaria. The surname Ghazi/Gazi is quite common in Romania, Bulgaria, Turkey, Slovakia, and other neighboring countries, which is perhaps the reason why it has been suggested that Gazey is originated from those countries.
So overall, I don't think the English Gazey from 1800s are of Arabic origin, and that the English Gazey are not related to the Arabic/Muslim/Southeastern Europe Gazey, if there are any with that spelling.