How To Stop A Leak In A Refrigerator Water Line
There a countless ways that the fridge line could have been installed. A photo of under the sink would greatly help, in particular showing the connection to the water supply.

Can you see where the fridge line connects to the water supply? Is it connected to the same supply that feeds the sink? If so, even with the isolation valves shut under the sink, there will still be pressure in the fridge line. If you open the sink's faucet, you can confirm that water is off, additionally this will relieve the pressure in the fridge's line.

Sometimes a small valve is installed on the fridge line under the sink. It could have been bumped open.

You'll need to determine how it is connected before you can fix it.