Holy cow, just found out the meaning of this song, written by Diamond himself. It's about wine! From Wikipedia:
Married to a catchy and dynamic melody and arrangement, the lyrics suggested to some a devotion to a woman of the night:
Oh, I love my Rosie child —
You got the way to make me happy.
You and me, we go in style ...
Cracklin' Rose, you're a store bought woman
You make me sing like a guitar hummin' ...
But in actuality, Cracklin' Rosie is a type of wine drunk by a native Canadian tribe that Diamond had visited in Canada. Apparently the tribe had more men than women. Cracklin' Rosie was the nickname they used for their homemade alcoholic brew, which the single men, who did not have dates, would sit around the fire and drink together.
In terms of wine itself, the title is seen to refer to (misspelled) rosé wine which is "crackling" - a U.S. term equivalent to pétillant or lightly sparkling. A Crackling Rosé is produced by, for example, Paul Masson Vineyards and Beckett's Flat.