Several EU leaders have criticised Turkey, amid a growing row over the Turkish government's attempts to hold rallies in European countries.
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.
On Sunday, Mr Rutte demanded Mr Erdogan apologise for likening the Dutch to "Nazi fascists".
"This country was bombed during the Second World War by Nazis. It's totally unacceptable to talk in this way."
The Netherlands would have to consider its response if Turkey continued on its current path, he added.
Meanwhile, German ministers also appeared to harden their rhetoric against Turkey.
Despite Chancellor Angela Merkel saying her government was not opposed to Turkish ministers attending rallies in Germany, as long as they are "duly announced", her interior minister said he was opposed to Turkish political gatherings in Germany.
"A Turkish campaign has no business being here in Germany," Interior Minister Thomas de Maiziere told local media.
Separately, Finance Minister Wolfgang Schaeuble said Turkey had "destroyed the basis for further progress in co-operation".
Reports say the owner of a venue in the Swedish capital, Stockholm, also cancelled a pro-Erdogan rally on Sunday that was to have been attended by Turkey's agriculture minister.
Sweden's foreign ministry said it was not involved in the decision and that the event could take place elsewhere.