Azar according to your source said:
After extensive contemplation and reflection, I have made the decision to exercise the right of violent retaliation that Allah has given me to the fullest extent to which I am capable at present.
I assume he is referring to the concept of Qisas in Islam. This concept is also cited by Daesh "terrorist" group as justification for retaliatory attacks against western nations.
According to Islamic jurisprudence it is prohibited to kill any women, children or elderly during battle. Islamic code of warfare according to the first Caliph, Abu Bakr RA (573 CE – 22 August 634 CE):
O people! I charge you with ten rules; learn them well!
Stop, O people, that I may give you ten rules for your guidance in the battlefield. Do not commit treachery or deviate from the right path. You must not mutilate dead bodies. Neither kill a child, nor a woman, nor an aged man. Bring no harm to the trees, nor burn them with fire, especially those which are fruitful. Slay not any of the enemy's flock, save for your food. You are likely to pass by people who have devoted their lives to monastic services; leave them alone. https://en.wikipedia.org/wiki/Islamic_military_jurisprudence#Prisoners_of_War
But Muslims are permitted to violate these rules if the enemy violates them. This permission is given in the following verse:
Yusuf Ali: The prohibited month for the prohibited month,- and so for all things prohibited,- there is the law of equality. If then any one transgresses the prohibition against you, Transgress ye likewise against him. But fear Allah, and know that Allah is with those who restrain themselves.
Sahih International: [Fighting in] the sacred month is for [aggression committed in] the sacred month, and for [all] violations is legal retribution. So whoever has assaulted you, then assault him in the same way that he has assaulted you. And fear Allah and know that Allah is with those who fear Him.
There is no definitive interpretation of this verse. Scholars of different times have interpreted it differently. It is upto those who are in authority to decide what constitutes a proportional response to enemies violation of ethics.
The retaliatory action is to be carried out by those who are in the authority not individuals. The use of extra judicial force is prohibited.
Azar as far as I know was a citizen of the USA which means that he has agreed to obey the laws of that specific nation. The prophet (saw) said regarding this:
“It is necessary upon a Muslim to listen to and obey the ruler, as long as one is not ordered to carry out a sin. If he is commanded to commit a sin, then there is no adherence and obedience.” (Sahih al-Bukhari, no. 2796 & Sunan Tirmidhi) http://www.daruliftaa.com/node/5852
It means that Muslims are to follow the rules of the land they live in if they are allowed to practice their religion properly. If they are forced to commit sins, then he/she shouldn't obey.
America sure has a lot of anti-Muslim sentiment which is reflected by the recent US elections but they don't (as far as I know) have any laws prohibiting the freedom to practice Islam. So, he does not have the right to practice violent retaliation since he is bound to follow the laws of the USA and since it's extra-judicial killing.