On that, a man from Al-Khazraj got up. Um Hassan, his cousin, was from his branch tribe, and he was Sad bin Ubada, chief of Al-Khazraj. On this, the two tribes of Al-Aus and Al Khazraj got so much excited that they were about to fight while Allah's Apostle was standing on the pulpit. Allah's Apostle kept on quietening them till they became silent and so did he. All that day I kept on weeping with my tears never ceasing, and I could never sleep.
All societies have laws and procedures how to deal with slander.
Muhammad did not call for a public trial that the slanderer be properly punished based on law agreed upon in the community. Muhammad went to the pulpit and asks for a volunteer among his companions to kill the man. According to this hadith, the order was not carried out in this case because the companions got into an internal fight and the original request of Muhammad might have been forgotten in the end. Internal quarrels saved his life. But the issue is whether Muhammad's approach to the problem is fitting a prophet who has to judge according to God's law and cannot just send a killer to execute whomever has annoyed him.