Juncker told the European Parliament that dialogue is the only way to tackle issues with Turkey, including the row over a German TV satirist who crudely insulted President Recep Tayyip Erdogan.
Berlin is weighing a Turkish request the satirist be prosecuted for slander in Germany but Chancellor Angela Merkel said Tuesday the issue is separate from the migrant deal.
"I cannot understand at all that a German ambassador is summoned for an admittedly difficult satirical song," Juncker told the lawmakers in the French city of Strasbourg.
"That does not bring Turkey closer to us, but will put us farther away from each other," said Juncker, the head of the European Commission, the executive arm of the 28-nation EU.
German prosecutors last week opened a preliminary probe against comedian Jan Boehmermann, 35, who accused Erdogan of having sex with goats and sheep while gleefully admitting he was flouting Germany's legal limits on free speech.
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The comedian was reacting to Ankara's decision to summon Germany's ambassador in protest last month over a previous satirical song broadcast on German TV which lampooned Erdogan in far tamer language.
Turkey's request to punish the satirist gave the affair a far broader diplomatic dimension and exposed Merkel to criticism she was kowtowing to Erdogan.
Juncker said Europe would stick to its guns.
"One thing is clear to me: no matter how important the work for refugees may be, our values on press freedom and fundamental values do not change," Juncker said.
But he added: "Dialogue will help us tackle these issues with Turkey."
The EU and Turkey sealed a deal in Brussels on March 18 to ease Europe's biggest refugee crisis since the end of World War II, under which all migrants arriving in Greece from Turkey will be turned back.
For its cooperation, Turkey won an acceleration of its long-stalled bid for EU membership, the doubling of refugee aid to six billion euros ($6.8 billion) and visa-free travel for its nationals to Europe's Schengen passport-free zone by June.
But EU officials said they would hold Turkey to EU press freedom and other standards in the accession talks.