What can be used to fill a wide gap on a curved hearth?
Although it may be a tad more complicated than plastic or vinyl, I would go with metal. Milled door sills are often produced in 80-100cm lengths (approx. 3'), either with screw holes or simply to be glued in place. You would have to cut it into short lengths with slightly angled ends; in essence you will be constructing a polygon instead of a circle. With a little searching, you will probably find a profile that combines well with the stone on one side, and carpet on the other. 

The rationale behind using metal is to create a supplementary barrier for any sparks that may fly with the glass open (e.g. when stirring the fire) and roll on the stone. I would also think some bronze would fit in rather well with your fireplace and poker holder, though that is of course a matter of personal taste.