Turkish President Recep Tayyip Erdogan threatened to send the refugees to Europe instead if it characterised the Turkish offensive as an occupation.
Turkish forces are stepping up air strikes and a ground offensive, as their incursion into Kurdish-held areas of northern Syria enters a second day. Turkey's military said it had seized designated targets. There are reports of heavy fighting in the central border region, and seven civilian deaths.
Tens of thousands of people are reported to be leaving their homes. The assault on Kurdish forces, key US allies, follows US President Donald Trump's decision to withdraw US troops. Turkey says it wants to create a "safe zone" on the border for many of the Syrian refugees on its territory.
"Hey European Union, wake up! Let me say it again - If you try to portray our operation as an invasion there, our task is simple. We will open the door and send you 3.6 million migrants," Erdogan said in a speech to Parlament.
Turkey launched a peace operation in Syria on Wednesday. It is targeting the Kurdish People's Protective Units (YPG), which Ankara regards as affiliated with the PKK, which has been banned in Turkey.
Erdogan said 109 "terrorists" were killed in the operation.
"We have a message for those forced to join the YPG line: if you leave immediately, we welcome you with open arms," the president said.
There are 3.6 million people living in Turkey who had to flee for the eight years of the Syrian war.
In 2016, Turkey signed an agreement with the EU to prevent refugees from moving to Europe in exchange for € 6 billion and visa-free travel.
Ankara, however, criticizes the EU for its slow transfer of money and inaction in resolving the refugee crisis.
"You've never been sincere," Erdogan said of the block. "Now they say they won't give us three billion euros. Have you ever kept any of our promises? No."
One of the objectives of the Ankara military operation is to establish a security zone in northern Syria where it can house about one million Syrian refugees on Turkish soil.
Turkish troops continued attacking Kurds in northern Syria on Thursday, the Turkish Defense Ministry said.
The second day's Turkish invasion has received widespread condemnation from the world. On Wednesday, locals tried to flee Northern Syria as quickly as possible in order to keep out of the war.
Kurdish forces and Syrian activists said on Thursday that Turkish troops had not yet made much progress despite the heavy bombing. However, there is no independent confirmation of these claims and the situation on the battlefield is difficult to assess.
According to the Turkish national news agency Anadolu, the Ankara backed Syrian rebels in Yabisa and Tel Fander villages. Their spokesman, Major Youssef Hammoud, called Yabisa the first freedom-winning village.
Turkey launched an onslaught against the Kurds on Wednesday with air strikes and cannon fire. Later, Turkish troops entered Syria. US troops who had previously supported the Kurds left the area.
Turkey had long threatened to launch an onslaught against Kurdish self-defense forces, which Ankara regards as being linked to the Turkish-banned Kurdistan Workers Party (PKK), which has been fighting the central power for decades.
The invasion came after US President Donald Trump's decision on Sunday to withdraw his soldiers from the area, leaving the Kurds vulnerable to the Turkish invasion.
Kurds, the only US allies in Syria to fight the extremist Islamic State (IS), stopped fighting the extremist group on Thursday to deal with the Turkish invasion, Kurdish and US officials said.
According to the Turkish Ministry of Defense, 181 targets have been bombed on the eastern bank of the River Euphrates by destroyers and cannons since the onset of the invasion.
Mustafa Bali, a spokesman for the Syrian Democratic Forces (SDF), said Kurdish fighters have repelled two attacks by Turkish ground forces, one near Tal Abyad and another at Ras al-Ain in the west. "They're not moving right now," he whispered Thursday.
"Our troops resisted an attempt by the Turkish Occupation Army to invade Tal Halaf and Aluki," the SDF said.
The Syrian Observatory for Human Rights reported that Turkish troops tried to penetrate several fronts in the shadow of air strikes and cannon fire, but no significant progress has been made so far.
On Thursday morning, the observatory announced that Turkish aircraft had launched new air strikes in the morning and Turkish troops had entered Beir Ashqi village near Tal Abyad.
According to sources, hostilities extend over 120-kilometer fronts in several sections.
According to the Observatory, 19 SDF fighters and eight civilians were killed in the first hours of the Turkish invasion.
A Turkish military operation in north-east Syria has forced more than 60,000 people to flee their homes in less than a day, the Syrian Observatory for Human Rights reported Thursday.
A large number of residents have left the border areas of Ras al-Ain, Tal Abyad and Derbasiyeh. Most of them headed east to the city of Hasakeh, the NGO said.
Shells fired from Syria injured 18 people in Turkey
Shells fired from Syria hit a Turkish border town on Thursday, killing 18 people.
According to a journalist from the Associated Press, two government buildings were hit in the border town of Akcaka in Sanliurfa province. Three people were seriously injured in one blast, a AFP journalist also said.
Smoke rose from several places in the city and the cannon fire has not stopped, AFP said. Families are evacuated and streets are empty.
In addition to Akcakale, Ceylanpınar was reported injured, according to Turkish media.
According to journalists, shells were fired across the border from Tal Abyad, a town in northeastern Syria.
Syrian Kurds have attacked at least five Turkish border towns with projectiles following the start of a Turkish troops' attack on Wednesday.
Residents were told to leave the border areas, stay indoors and stay alert.
Kurdish leaders on Thursday urged European countries to call their ambassadors home from Ankara to protest against the Turkish invasion.
A delegation of the Syrian Democratic Forces (SDF) political wing traveled to Brussels to influence the European Union to take steps to punish Turkey.
The EU urges Turkey to stop the invasion, but has done nothing. Community foreign ministers will discuss the situation at Monday's meeting.
"We want an immediate response, these attacks must be stopped quickly. Airspace must be closed to Turkish aircraft so that air strikes can be stopped," SDC chief Ilham Ahmed told reporters in Brussels.
"All European countries should freeze relations by immediately bringing ambassadors out of Turkey."
The SDF was an important ally of the United States in the fight against the ISIS extremist group, but Ankara regards Kurdish fighters as terrorists like Kurdish rebels in Turkey.
Ahmed resented the withdrawal of US troops from the Syrian-Turkish border, which made it possible for Turkish troops to invade.
"If American troops were present, Turkey would not attack us, but unfortunately, America's retreat put us at risk," he said. Ahmed reminded Washington of the Kurdish sacrifice in the fight against IS.
"We want them to take into account the high price we paid - 11,000 martyrs - and stay with us."