A homeowner called and said his new faucet wouldn’t shut off the cold water. Now that’s strange. He was looking under the sink to try to figure it out. The problem has to be related to the faucet and not under the sink however. This was a two handled faucet and so the problem is isolated to the cold water side.
It might be as easy as repositioning or tightening the handle depending on the style. Sometimes if the handle is not engaging the stem it either won’t turn it all the way on or off, or if it is not correctly aligned, it may hit an integral stop before it is meant to. They will show itself as either low pressure or in your case, the water won’t shut off. You can remove the handle, turn the stem by hand until the water shuts off, and then correctly reposition the handle and tighten.
In the event that is not the problem, you may need to replace the stem and or seat (although this would be rare in a new faucet). You might call the manufacturer of the faucet and they can send out a new free part….or just take the faucet back to the retailer for a new one.