"action of turning, bending, or forming into ringlets," mid-15c., verbal noun from curl (v.). Curling-iron "rod of iron to be used hot for curling the hair" is from 1630s.
The game played with stones on ice, originally Scottish, is so-called by 1610s, but the sense connection is obscure. "The name appears to describe the motion given to the stone" [OED]. Evidence of the sport dates to the early 16c. in Scotland; written accounts of the game date to the 1540s. A similar game is described from c. 1600 in Flanders.