Saddle valve on water line running to fridge won't shut off
If water is still coming out after turning off the valve, the valve needs to be repaired or replaced. It's possible that there is some debris or hard water deposit in the valve that is preventing it from sealing and cleaning the valve will do the trick, but a new saddle valve is so inexpensive ($15 or less) it really isn't worth any time diagnosing it; I would just replace it.

The danger is that taking the saddle valve off of the pipe to inspect or replace it might cause a leak all on its own; you may introduce a new leak between the valve and the pipe when you reassemble it.

A bigger concern in my opinion is that your main water shutoff has the same problem. If your home is going to sit vacant while on the market and you are planning to turn the water off or if you are trying to get your house in tip-top shape to sell, that's what I would hire a plumber to fix. Maybe ask him to put a new saddle valve on in the kitchen while he's there.

If you're not interested in investing money in the house or not worried about needing to have the water absolutely off after you move out, as @JPhi1618 mentioned in a comment on your question: you could do a band-aid repair by putting a cap on the end of the line to prevent damage to the flooring and let the next owner deal with it.