The Turkish president has warned that "millions" of migrants and refugees will soon head towards Europe.
Recep Tayyip Erdogan was speaking after announcing Turkey could no longer enforce a 2016 deal with the EU to prevent migrants entering Europe.
Mr Erdogan said Turkey could not cope with a new wave of refugees after an escalation of the Syrian conflict.
A young boy died when a boat capsized off the Greek island of Lesbos on Monday, Greek police said.
It was the first reported fatality since Turkey opened its border last week.
Meanwhile, a Turkish official accused the Greek authorities of killing a Syrian man who was trying to breach the border on Monday. Athens denied the allegation.
An unverified video showed a man on the ground with blood on his neck.
Nearly a million Syrians have fled to the Syrian-Turkish border since December, amid heavy fighting in the Idlib region between Turkish-backed rebels and Syrian government forces.
Turkey is already hosting 3.7 million Syrian refugees, as well as migrants from other countries such as Afghanistan - but previously stopped them from leaving for Europe.
In a televised speech, Mr Erdogan said the EU had provided no help for resettling Syrian refugees in "safe zones" inside Syria.
After Turkey opened its doors for migrants to leave its territory for Europe last week, he said, "hundreds of thousands have crossed, soon it will reach millions".
The Turkish leader gave no evidence for his figures, while Greece said about 1,000 migrants had reached its eastern Aegean islands from Turkey since Sunday morning.
In addition, Greece said it had stopped almost 10,000 migrants from crossing its land border in 24 hours. Some migrants tossed stones and metal bars when stopped, and Greek border guards fired tear gas.
What is the 2016 EU-Turkey deal?
- It was proposed as a solution to the migrant crisis in which almost one million refugees and migrants arrived in the EU in 2015 and thousands died in mass drownings
- On 20 March 2016 the EU and Turkey made a deal that saw Syrian refugees who arrived on Greek islands sent back to Turkey. In return Turkey received €6bn (£5.2bn; $6.7bn) in EU aid for migrants and refugees
- For every Syrian person removed from Greece to Turkey, another would be resettled from Turkey to the EU
- The EU agreed to work towards lifting visa requirements for Turkish citizens by the end of June 2016
- After the deal was agreed, the number of migrants arriving in Greece declined sharply
Besides Syrians, there are Afghans and West Africans among the migrants at the border.
Europe's top human rights watchdog, the Council of Europe, condemned Greece and Turkey's actions, calling the situation an "unprecedented humanitarian crisis".
"Everything must be done to de-escalate violence in the border region, including by ensuring that law enforcement authorities refrain from using excessive force," it said.
EU Council President Charles Michel and EU Commission President Ursula von der Leyen will go to the border with Greek Prime Minister Kyriakos Mitsotakis on Tuesday.
"Either we bring these people to a dignified life in their own land, or everyone will get their share of this burden. Now the unilateral sacrifice period is over," Mr Erdogan said.
In a statement on Monday, the Greek government said, "Turkey, instead of curbing the traffickers' channels of migrants and refugees, has itself turned into a trafficker."
Greece has blocked any new asylum applications for the next month, because of what it calls the "coordinated and massive nature" of illegal migration from Turkey.
"This relocation of people has nothing to do with international law regarding the right of asylum, which concerns only individual cases," the statement said.
The United Nations Human Rights Council said that Greece had no legal justification for suspending asylum procedures.
Frontex says it is examining how best to help Greece with a rapid deployment of EU border guards.
Frontex already has nearly 400 personnel in and around the Greek islands, 60 officers in Bulgaria and a smaller contingent in the Evros region, at the Greek-Turkish border.