Suppose the live wire comes into contact with the Earth wire. There would not be a complete circuit for the current to travel across. Therefore, why would there be a short circuit and why would the fuse melt? Can someone guide me through the process of what happens if the live wire touches the Earth wire (and not specifically the metal casing of an appliance)?
The "earth wire" is attached to ground; typically via a cold water pipe, which is in actual contact with damp earth.
If the ground connection is good, it provides a nearly infinite capacity to accept charge, all the time maintaining the same relative voltage level.
Because charge flows according to the difference in voltage, and along low-resistance paths, the combination of the "wire" with the "earth" results in all of the current flowing through this path. As the current exceeds the rated capacity of a fuse it melts/blows, or with a circuit breaker, it pops.
This opens the circuit, and the current flow stops.
There would not be a complete circuit for the current to travel across.
Actually, there is. The specifics depend on your jurisdiction. But in the US, the ground wire is bonded to the service neutral at the panel. So there is a very low resistance path from the hot, through the short, the ground wire, back to the neutral and then back to the transformer. This low resistance path allows sufficient current to flow that the circuit breaker (or fuse) will trip.
If the ground wire only went to ground, there is still a circuit because the power station also uses the ground as a voltage reference. But this circuit through the earth would be much higher resistance. The current that could flow this way might be insufficient to blow a fuse or trip a breaker.