Answer:
No. Natural gas is mostly methane, an odorless gas. The smell you detect is a compound called methyl mercaptan, a mostly natural substance found in the blood, brain, and some other animal and plant tissues. It's present in feces, and occurs naturally in certain foods such as cabbage, some types of nuts and cheese. It is the main chemical responsible for bad breath and the smell of farts. The human sense of smell is very well adapted to detect it's odor in minute quantities, as low as 0.002 ppm. Though some natural gas wells contain methyl mercaptan, it's mostly added to the gas by the distributors as a means of detecting leaks.
The odor most commonly associated with acetone is nail polish remover, which smells nothing at all like farts!