Is Antonio justify of killing Vindice?

On balance, one could argue that Antonio is indeed justified in ordering the execution of Vindice. Look at it this way: throughout the course of the play Vindice has shown himself capable of numerous acts of double-dealing, duplicity and murder. Though initially a sympathetic figure on account of his desire to gain revenge on the wicked Duke, he's become considerably less so as the story's progressed. Vindice has found himself embroiled in the dark arts of court intrigue, becoming every bit as morally corrupt as the louche, back-stabbing aristocrats who hang around the Duke's palace.
This makes Vindice a dangerous man to have around. So when Antonio becomes Duke, an instinct for self-preservation kicks in. Having gone to so much trouble to ascend to the throne, Antonio's not about to give up his new-won power so easily. Vindice is a threat to that power; and what happened to the old Duke can just as easily happen to Antonio. So Antonio's decision to have Vindice executed could be justified as a preemptive act of self-defense.