Without Newton's third law of motion, what would an object sitting on a table do?

Without Newton's third law of motion, an object sitting on a table would continue sitting there, motionless, just we always see it do, and just as objects used to do before Newton ever developed his laws of motion.

Getting the new law didn't change anything. Just like everything else in Science, the new law was just a new description of how Nature works. But it didn't change anything, because Nature doesn't care whether we understand how it works or not.

Newton's third law: If an object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on object A. This law represents a certain symmetry in nature: forces always occur in pairs, and one body cannot exert a force on another without experiencing a force itself.