The statement came on the 40th UN Palestinian Solidarity Day, which is being marked by pro-Palestinian campaigners but receiving little attention from UN member states' governments.
The healthcare system on the Gaza Strip, which subject to a tight Israeli blockade, is being overwhelmed by the hundreds of cases cause by Israeli fire during demonstrations at the border, MSF warned in a statement on Thursday.
The protesters are demanding the right to return to homes their families lived in before the creation of Israel in 1948.
According to the Gazan health ministry, about 6,000 Palestinians have been shot by Israeli forces during the demonstrations, during which the Israeli soldiers are sometimes targeted by snipers and slingshots.
"This many patients would overstretch the best healthcare systems in the world," Marie-Elisabeth Ingres, MSF's Palestinian Territories chief, said in the statement. "In Gaza, it is a crushing blow."
MSF called on Israel to allow the injured out of the strip for treatment and for governments worldwide to officer medical facilities to them.
If that does not happen "thousands of patients will be left to deal with terrible injuries, with many permanently disabled and dependent on their families", it said.
Three Gaza wars
Israel has launched three wars against the Gaza Strip, which is controlled by the Islamist Hamas movement.
Many of its two million residents are refugees from what is now Israel.
At least 235 Gazans have been killed since 30 March, mostly by Israeli fire during border clashes but some by air and tank strikes.
Two Israeli soldiers have been killed over the same period, one by a Palestinian sniper and another during a botched Israeli special forces operation inside the strip.
Palestinian solidarity day
The statement appeared on 29 November, the day the UN General Assembly declared the International Day of Solidarity with the Palestinian People 30 years ago.
The resolution called on member states to "give the widest support and publicity to the observance of the day", the UN's website declares, but the appeal appears to have fallen on deaf ears so far as most governments are concerned.
Pro-Palestinian campaigners are marking the day, however, and they are sharply critical of the policies of countries like France.
"European countries play a key role since they are the first partners for trade with Israel," Olivia Zémor of the French group EuroPalestine told RFI. "They consider Israel a European country and therefore grant it enormous privileges such as not having to pay any taxes on the products they export to Europe."
France has a contradictory policy, Zémor claims, in particular in relation to Israeli settlements in the Palestinian Territories.
"Our government says that settlements are illegal, that occupied Palestinian Territories do not belong to Israel, yet they let Israel export their products from settlements without saying anything and they are exempt from taxes.”
BDS activists prosecuted
A number of activists, including Zémor herself, have been prosecuted for calling for boycotts of Israeli products or those from settlements.
The case against her was dismissed, as have several others, but there have been some guilty verdicts.
Pro-Israel groups have brought the cases, citing a 2010 circular by then justice minister Michèle Alliot-Marie that called on prosecutors to take legal action against calls to boycott Israel on the grounds of incitement to racial hatred.
And some local authorities, including the Paris city council, have passed resolutions condemning the Boycott, Divestment, Sanctions (BDS) movement and refusing to let rooms to it.
France's highest court has asked the European Court of Justice to rule on calls to boycott products from settlements.
Zémor accuses successive French governments of being too cosy with Israel's leaders, including current Prime Minister Benjamin Netanyahu, who heads a troubled right-wing coalition.
“The French governments, not only [Emmanuel] Macron but [François] Hollande and [Nicolas] Sarkozy, were rolling out the red carpet regularly for Netanyahu and at the same time read reports by Unicef, by Amnesty International and Israeli NGOs stating that there is child detention and the torture of Palestinian children,” she comments.
EuroPalestine planned a meeting in Paris bookshop and a "surprise" event on Thursday evening to mark the solidarity day.
Israel's first prime minister David Ben Gurion used to say of the Palestinians that "the old will die and the young will forget", she says. "The surprise is that maybe a lot of them died but there are many young people in Palestine that did not forget and they’re fighting for their rights.”