At least 22 people died in the air strike, including a dozen doctors and seven adult patients and three children, according to Doctors Without Borders, an non-governmental organization known internationally by its French name Medecins Sans Frontieres or its acronym MSF.
"Today, we say enough. Even war has rules," said MSF Executive Director Jason Cone said today during a news conference in New York.
His comments echoed those of MSF's international president Dr. Joanne Liu during a news conference in Geneva this morning.
"In Kunduz, our patients burned in their beds. MSF doctors, nurses and other staff were killed as they worked. Our colleagues had to operate on each other. One of our doctors died on an improvised operating table -- an office desk -- while his colleagues tried to save his life," Liu said.
What MSF Wants to Have Happen Now
Earlier today in Geneva, Liu called for the International Humanitarian Fact-Finding Commission to investigate the possible violations of international humanitarian law.
This would be the first time that the commission has ever been used. It was initially formed in 1991 the but it has never launched a fact-finding investigation.
"It requires one of the 76 signatory states to sponsor an inquiry. Governments up to now have been too polite or afraid to set a precedent. The tool exists and it is time it is activated," Liu said today.
Cone called for President Obama to consent to the mission, saying that it would send a "strong" statement of support for the U.S. to be the first country to sign on to the commission's investigation.
The main reason for involving the never-before-used agency is because the MSF is calling for it to be "investigated independently and impartially," which they believe would not be possible if either government involved was at the helm of a probe.
The questions raised about the partiality of the players involved was raised "given the inconsistencies in the U.S. and Afghan accounts of what happened over recent days," Liu said.
"It is unacceptable that states hide behind ‘gentlemen’s agreements’ and in doing so create a free-for-all and an environment of impunity. It is unacceptable that the bombing of a hospital and the killing of staff and patients can be dismissed as collateral damage or brushed aside as a mistake," she said.
What the US Has Said
Defense Secretary Ash Carter and Gen. John Campbell, the commander of the Resolute Support training mission in Afghanistan, have had to walk back statements made over the course of the four days since the Saturday air strike, only to walk back those statements again. At various points, the statements have differed over whether or not U.S. troops were on the ground near the hospital, if there were Taliban fighters directly engaging with the U.S. forces on the ground, and if it was the Afghan or American forces who called in the strike.
Campbell appeared before the Senate Armed Services Committee on Tuesday and said that though the airstrikes that struck the MSF hospital in Kunduz were requested by Afghan forces, they were approved by the American chain of command.
No U.S. official had apologized until President Obama called Liu this afternoon to "apologize and express his condolences for the MSF staff and patients who were killed and injured when a US military airstrike mistakenly struck an MSF field hospital in Kunduz, Afghanistan over the weekend," White House spokesman Josh Earnest said at a press briefing.
"We received President Obama's apology today for the attack against our trauma hospital in Afghanistan," Liu said in a statement. "However, we reiterate our ask that the U.S. government consent to an independent investigation led by the International Humanitarian Fact-Finding Commission to establish what happened in Kunduz, how it happened, and why it happened."
"On Saturday morning, our forces provided close air support to Afghan forces at their request. To be clear, the decision to provide aerial fires was a U.S. decision made within the U.S. chain of command. A hospital was mistakenly struck. We would never intentionally target a protected medical facility," he said.
What Happens Now
There are three investigations under way.
On the American front, the ongoing government-backed internal military investigation in Afghanistan headed by Brigadier Gen. Richard Kim will continue as well as a complete review of operating procedures.
Gen. Campbell said Tuesday that the "entire force" has to be re-trained and review the rules of engagement that the U.S. military operates under, saying that it was ordered "to prevent any future incidences of this nature."
The rules of engagement for American and Afghan troops in Afghanistan to request airstrikes are that they can be called in for counter-terrorism strikes, force protection, or in extremis situations to prevent casualties.
ABC News' Luis Martinez and Kirit Radia contributed to this report.