Mr Khan also underlined the need for humanitarian aid to get into Gaza, with access to food, water and medical supplies a basic right for civilians "without further delay, and at pace and at scale". He also urged Hamas not to divert or misuse the supplies when they did arrive, after accusations by Israel that its members were looting trucks and stealing aid, stopping it from reaching the wider population.
Israel isn't a member of the ICC and it says the court has no jurisdiction over it. But the court has ruled that since 2015 it does in fact have jurisdiction over the West Bank, East Jerusalem and Gaza after the Palestinians ratified its founding treaty, the Rome Statute, as the State of Palestine.
No Western-style democracy has had an ICC arrest warrant issued for its leader before. If it happens to Mr Netanyahu, he'll be the first. Israelis have particular concerns about the stigma and potential isolation that would come with it.
Michael Oren was Israel's ambassador to the United States between 2009 and 2013. He points to the recent case brought by South Africa at the International Court of Justice (ICJ), which accused Israel of genocide against the Palestinian population in Gaza - an accusation Israel vehemently rejects.
In January, the court issued an interim ruling that ordered Israel to take steps to prevent genocidal acts in Gaza, but stopped short of telling it to stop its military offensive.
"Israel emerged from those proceedings relatively unscathed, but the fact that the proceedings were held at all meant that Israel had lost the battle," he told the BBC.
"They should not have been held at all, and certainly it's a blow to our international standing and our security. Because a country whose leaders are accused of presiding over war crimes, that country is more vulnerable than a country that isn't."
Sir Geoffrey Nice KC, another British barrister, led the prosecution of the former Serbian president Slobodan Milosevic for war crimes at the International Criminal Tribunal for the former Yugoslavia (ICTY).
He points out that the actions of both sides in this conflict are within the ICC's focus.
"Any investigation will not just be the Israeli Defense Forces and political and military leadership but also Hamas," he told the BBC.
"It's to be hoped that alleged Hamas fighters and leaders are not all shot on sight, that they are detained and made available for trial by an international tribunal."
He also addressed suggestions that the ICC could come under political pressure to reconsider its decision-making.
"Governments around the world, particularly large and powerful governments, are there to protect their country's interests.
"So if they see a trial coming along, or an investigation... that would cause them as a country great reputational damage, and if they reckon they can interfere with the trial or the investigative or trial processes to save them from that reputational damage, then they'll do that, because they are acting in the interests of their country."