From both an ethical and a legal standpoint, the United States government can only kill Americans in a rather limited set of circumstances. These, obviously enough, typically involve criminal activity on the part of a citizen that justifies such a killing.
Interestingly, the government also has the legal right to assassinate American citizens who are involved in terrorism, provided that the proper procedures are followed. Not surprisingly, this might strike some as rather disturbing and perhaps unethical. After all, the good guys are not supposed to assassinate people and certainly not fellow Americans. However, there seems to be a moral basis for Americans killing American terrorists.
One case is rather obvious: when an American citizen is willingly part of a terrorist group and is killed in the course of operations against that group. For example, if an American joins a group of terrorists attacking a Marine unit in Afghanistan and he is killed by the soldiers, then that would seem to be morally acceptable under the usual moral rules governing war. After all, when a citizen willingly joins enemy forces as a combatant, then he falls under those moral conditions.
Of course, it becomes more controversial when an American is specifically targeted for assassination. After all, as assassination is not the same sort of thing as killing in the course of a military engagement. In some cases, assassination could be seen as not being an actual assassination, but as a legitimate part of law enforcement. For example, when a police sniper takes out someone who is trying to kill a hostage, that is not assassination. So, if an American citizen was involved in an act of terror and had to be killed to protect others, that would be morally on par with the legitimate use of force on the part of the police.
Where it becomes very controversial is when it is a true assassination. That is, intentionally seeking out a specific person and killing him so as to kill him, rather than as part of dealing with a specific situation (such as a hostage situation) that requires force. Or, to be blunt about it, committing murder to achieve political goals. For example, targeting an American at a terrorist training camp with a missile strike.
It is tempting to justify such assassinations on the grounds that by killing the American terrorist the assassin is protecting America-just as the police officer who kills a hostage taker before he can kill his hostage. However, there is an important distinction: if the police went around killing criminals while they were merely planning or preparing for crimes, that would be both illegal and immoral. Likewise, assassinating an American terrorist in similar circumstances would also seem to be wrong. After all, he is still a citizen and thus still entitled to the due process of the law. Interestingly, it seems that arguments that could be given for assassinating American terrorists (or other political enemies) in such circumstances would also justify the proactive assassination of dangerous criminals. However, some folks might see this just fine-after all, they might say, real Americans cannot be bothered with all that legal stuff like due process and trials.
However, a citizen who becomes a terrorist or a vicious criminal is still a citizen and hence assassination would not be a morally acceptable option. If we take the notion that people have a right to life seriously, that means that assassination would be quite out. That said, it might still be ethical to kill such a person, but that would require the proper circumstances and due process (which, in some cases, might be a shot to the face during a firefight).