Normal function is to put ice maker arm in UP position. It could be a switch on newer fridges. This would tell fridge valve not to open/send water.
If water cannot be stopped, needs to be turned off mechanically. Pull fridge from wall, follow copper or plastic tubing to water supply source and hopefully there is a water shutoff valve to throw. This may turn off water in the fridge door too if that's a feature. If no shutoff, that leaves another shutoff somewhere else, possible the whole house shutoff valve so a repair can be made.
Hopefully we wil have a repair person today. If not... we will try and pull out frig as I am tired of getting up every hour all night to dump water! Both being over 70 makes moving this big frig a problem but worth the try. I did empty the freezer this AM so on the right track. Thanks