The bottom line is al-Awlaki was a TERRORIST. He put his own children in harm's way, as terrorists always do. They use children as shields and then whine when their shields are harmed. He is responsible for their deaths and NOBODY ELSE.
Down and dirty the above may be true. As using non combatants as human shields is nothing new under the sun for radical Islamism.
However, killing any non combatants is prohibited by the Laws of Armed Conflict (LOAC).
Unless...Unless not engaging the target presents an imminent threat to other more populated areas with non combatants or combatant forces. For example, if a terrorist was assembling a chemical agent to kill thousands or millions, the justification of destroying the target area becomes an imminent threat, thus the human shields dying although a tragedy would be deemed acceptable collateral damage. The necessity of the attack to save many more lives outweighs the few human shields. Yes kind of a Kobiashi Maru situation.
Another situation is where a terrorist group holes up in a hospital or apartment building. Both structures alone are protected facilities by LOAC. However, if the combatant terrorists use the sites to directly engage opposing forces, by LOAC the opposing or pursuing forces can engage as right of self protection. That building for THAT specific engagement loses its protected status.
Ok legalese aside, leaders have to make moral decisions within the LOAC.
In my chemical assembly example, a commander must justify the urgency of the situation outweighs the death of human shield non combatants. This will be based on intelligence collection. Very detailed pattern of life intelligence, with detailed corroboration of intelligence on the urgency of an upcoming attack. Very difficult to do and we all know the history of such situations going very badly. We've had some complete disasters.
For the protected facility example, I emphasized the importance of the current situation. If US helicopters are flying around looking for the enemy and then take direct effective fire from the rooftop of a hospital; and the crews take casualties or damage, or are put in imminent danger, by LOAC, the crews can proportionally return fire on the hospital roof top.
Again for that particular fire fight and any pursuit.
Now proportionality is key. The crew can return fire to handle the threat. Not call in a B1 to level the hospital. That would be disproportionate and illegal.
Also, after the exchange at the hospital ceases, you cannot use that one incident to justify a daily bombing campaign on the hospital. You can send ground troops to clear the hospital to safeguard non combatants.
That's the down and dirty.
If al-Awlaki presented a clear and imminent threat to the greater populace in the specific time and space of the attacks on him, LOAC would support the attacks.
However, I call such attacks into question. Why? I used to staff Kinetic Strike packages in Iraq for 3 and 4 star generals. I was the one who briefed them directly on each packet, describe the pattern of life analysis and the projected non-combatant casualties. Each and every time when non-combatants were involved, my 3 star General boss would pick up the phone and tell the 2 star to use direct action on the ground instead of bombing a house or mosque with non combatants present. Every time.