President Recep Tayyip Erdogan accused Germany and the Netherlands of "Nazism" after officials blocked rallies there.
Dutch PM Mark Rutte called his comments "unacceptable", while Germany's foreign minister said he hoped Turkey would "return to its senses".
Denmark's leader has also postponed a planned meeting with Mr Erdogan.
Prime Minister Lars Lokke Rasmussen said he was concerned that "democratic principles are under great pressure" in Turkey.
He added that he had postponed the meeting because: "With the current Turkish attacks on Holland the meeting cannot be seen separated from that."
The rallies aim to encourage a large number of Turks living in Europe to vote yes in a referendum expanding the president's powers.
However, planned rallies in Germany, Austria and the Netherlands were blocked after officials cited security concerns or said the rallies could stoke tensions.
A gathering in France however went ahead after local officials said it did not pose a threat.
Ties between the Turkish and Dutch leaders became particularly strained at the weekend after two Turkish ministers were barred from addressing rallies in Rotterdam, with one of them escorted to the German border.
Mr Erdogan likened the Netherlands to "a banana republic", demanded international organisations impose sanctions on the Netherlands, and accused countries in the West of "Islamophobia".
"I have said that I had thought that Nazism was over, but I was wrong. Nazism is alive in the West," he added.