Erdogan lashed out over the weekend after rallies in the German city of Cologne and the town of Gaggenau in support of his government were canceled due to concerns about overcrowding.
In a video on the website of Anadolu
, the Turkish state news agency, Erdogan said: "I thought that Nazism was over in Germany, but it turns out that it is still going on. It is still going on, it is clear."
The President added: "My brothers; now they think that Tayyip Erdogan will come to Germany. I will come tomorrow if I want to. I will come and when they won't let me enter or don't let me talk, I will set the whole world in an uproar."
On Monday, Merkel hit back, accusing Erdogan of minimizing the atrocities committed by Hitler's regime.
"One cannot even comment on such utterances, they are not justifiable," she said, speaking at a business forum in Berlin.
"Comparisons with Nazis always just lead to one thing -- to belittle those crimes."
Earlier, Steffen Seibert, Merkel's spokesman, called for "cool heads to prevail," describing the situation as "a deep-seated disagreement which makes us very concerned."
"We reject the policy identification of the democratic Germany with that of National Socialism. National Socialist comparisons are absurd and not acceptable," he said at a press conference.
"The Federal Republic of Germany and Turkey have a close economic relationship; they are also partners in the fight against terrorism."
The rallies, at which Turkish government ministers were due to speak, were being held to drum up support for the upcoming referendum vote on the Turkish constitution on April 16.
The sweeping constitutional changes, introduced by Erdogan's ruling Justice and Development Party (AKP), would grant new powers to the President and transform the way the country is governed.
The world's largest Turkish diaspora is found in Germany, where about 3 million people with Turkish ethnicity reside.
The dispute over the rallies marks the widening of a diplomatic rift between the two countries.
Last Monday, a Turkish court arrested Deniz Yucel
, a correspondent with the German newspaper Die Welt, on charges of terror propaganda and inciting hatred and enmity.
Yucel -- a dual German-Turkish citizen -- had reported on leaked emails from Energy Minister Berat Albayrak, which were hacked by leftist collective RedHack and made available on WikiLeaks. He remained in custody.
Diplomatic spat 'quite serious'
The spat comes at a time when political relations between Turkey and Germany are at a low ebb, according to Enes Bayrakli, an assistant professor of political science at the Turkish German University in Istanbul.
"This is quite serious," Bayrakli told CNN. "What we have seen in the last couple of years is a deterioration in political relations between Turkey and Germany. We have good economic relations but not on the political side, and it's getting worse each year."
Bayrakli said Ankara feels Germany is too soft on Gulenists and members of the PKK -- a Kurdish separatist group that Turkey has been battling for decades -- living in Germany.
And he said German-Turkish relations have worsened since the attempted coup against Erdogan last July.
In the aftermath of the coup
, German authorities banned Erdogan from addressing a rally of Turks in Cologne, via video link, who were demonstrating against the coup attempt.
"All these developments are rather worrying," Bayrakli told CNN.