Fat came before diet drinks, but I seem to remember seeing a paper relating carbonated drinks to a desire to eat, so you may be right.
In past days, a man with a big belly spoke of it as his 'corporation'- as he drank his brandy and soda before dinner.

A song I'll have to dig out of my notes began, if I remember:
There lived a fat old lady
In Boston she did dwell
She loved her old man dearly
But another one twice as well