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The United States says it is conducting a "full investigation," but the medical aid group, also known as Medecins Sans Frontieres, or MSF -- which lost 12 staff members in the attack -- has called for an independent inquiry "under the clear presumption that a war crime has been committed."
Doctors without Borders is asking for an investigation to be carried out by a never-before-used commission: the International Humanitarian Fact-Finding Commission.
It's an open question as to whether an inquiry could be initiated, as the commission's website says it requires the consent of the parties involved, and neither the United States, Afghanistan, nor France recognizes the commission.
CNN asked Steven Kay, who defended Kenyan President Uhuru Kenyatta against charges of crimes against humanity at the International Criminal Court, and Anthony Dworkin, co-editor of "Crimes of War: What the Public Should Know," about the case.
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What constitutes a war crime?
In general terms, a war crime may be committed when there is an attack on a civilian population during an armed conflict, Kay says.
There is an extensive body of law that regulates military action during conflict.
The aim, says Dworkin, is to draw a balance between what armed forces are justified to do out of military necessity -- which includes causing collateral damage to civilians -- and the principles of humanity.
"Hospitals enjoy a special protected status under international humanitarian law. So, to attack a hospital or medical facility, whether it is a civilian or military installation, is a crime," says Kay.
However, if the hospital is used to support military operations for nonmedical purposes, then it loses its special protection status and it can be attacked by the opposition forces.
But the law requires that the attack be proportionate to the threat and risk involved.
Can the hospital bombing be considered a war crime?
Doctors without Borders says the hospital was bombed for more than half an hour after it notified NATO and U.S. officials that the facility was under attack.
In this case, according to Kay, the hospital bombing would be justified, and not considered a war crime, only if there was a clear, imminent or ongoing attack at the hospital.
Kay notes that there have been instances of medical installations being used as cover for forces to launch attacks.
Kay cites examples including allegations that during the last three decades of the Sri Lankan civil war, the separatist Tamil Tigers based fighters and used force at protected sites like civil hospitals. A U.N. panel found credible allegations of war crimes committed by both sides during the final stages of the country's civil war.
Dworkin makes the point that hospitals can be attacked by mistake.
In comments Monday, Gen. John Campbell, the commander of U.S. forces in Afghanistan, described the hospital deaths as an error, saying "several innocent civilians were accidentally struck" in an operation targeting Taliban fighters.
Dworkin says he believes that in this case, the United States would not be found guilty of committing a war crime because although armies are forbidden from conducting indiscriminate attacks, "the laws do recognize that mistakes happen in the fog of war."
But it would come down to a matter of evidence and establishing exactly what happened.
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What proof is required to prosecute a war crime?
"Beyond reasonable doubt -- the criminal standard of proof," says Kay.
He adds that cases like this are generally quite easy because state military operation are often well documented.
"Satellite photographs, drones ... all this is recorded in live time," he says. "Military operations don't happen in a garden shed. Hundreds of people (are) involved in planning an operation and monitoring, because they are monitoring a plane flying in the sky that is worth $100 million."
What happens next?
Armed forces have a duty to investigate the conduct of their own members because if there has been breach of humanitarian law, they have the first responsibility to punish those responsible, says Kay.
It doesn't automatically escalate to the international level, but there is a duty on the state involved to investigate and discipline, he says. Failure to investigate and discipline adequately would then bring scrutiny at a higher level, he adds.
Can the U.S. be prosecuted at the ICC for war crimes?
The International Criminal Court is the first permanent court established to bring to justice to those who commit the most serious crimes of concern to the international community.
The ICC prosecutes individuals, not countries, and in theory the person who gives the command can be held responsible for the crime, just as the subordinate who executed the order can be, says Dworkin.
However, the jurisdiction of the ICC extends only to states that have ratified the treaty that created the court. The United States is not a state party to the ICC treaty, or so-called Rome Statute, its citizens cannot be tried at the International Criminal Court.
In this particular case, says Dworkin, Afghanistan is a signatory to the ICC, so the court would have jurisdiction over any crimes committed on its territory.