Criminal law in each state is different so that without knowing which state it is not possible to argue one way or the other. Under common law, a home owner can use deadly force to protect his or her house but only when he is encountering life threatening force. In this case, the thief committed a burglary, which is a felony. However, burglary is not a robbery which could give the homeowner rights to use deadly force. There is no information on whether the thief actually took anything and the thief is not using force or threatening to use force, as the owner believed to be, such that it does not amount to a robbery. Under the circumstances the homeowner committed murder, most likely 2nd degree. Involuntary manslaughter is also another likely charge.
Under some state laws, a home owner can use deadly force to prevent a fleeing felon. But under Tennessee v. Garner, the Supreme Court made it clear that only when felon threatened with serious bodily injury or death, no deadly weapon can be used. Other state laws will allow protection of a home but in this case the owner did not aim to protect his house but to get revenge, more likely.
Most Chinese believe if some guys invaded their home the owner has the right to kill. This is deadly wrong. ONLY under one situation you can kill, that is, when you have right for self-defense. That hurdle is extremely high.