No you cannot. A single handle faucet like yours has a mixing valve inside it that connects the cold and hot side with each other. In other words, there is a point where the hot and cold water mix together.
The cold water will travel back through this point and out the bottom of the disconnected hot side. Big mess. Even if you keep the faucet all the way to the cold side, some mixing could occur. Also you’re now relying on every possible user to not turn the faucet even slightly away from the maximum cold setting. Simple human error will cause water to come gushing out under the sink.
Just cap off the hot side. If you leave the hot side open, you’re asking for trouble. As Ecnerwal said in the above comment, one good solution is to add a tee fitting to the cold supply line and bring cold water to both the hot and cold side of the faucet. Doing so will ensure that there’s always water flowing no matter which way the faucet is turned.