No! While curling is played on ice it is not a sport which ever uses ice skates. We wear clean rubber-soled shoes on the ice. Any shoes that have a grippy, fairly flat, rubber sole will do. Shoes designed for cold weather and slippery surfaces tend to do better.
Frequent curlers will purchase curling shoes which provide better traction on the ice, and have a built in slider.
We do not allow anyone to curl with open-toed shoes or high heels.