Turkey reacted angrily when Germany's constitutional court banned Mr Erdogan from appearing by video link at Sunday's demonstration in Cologne, accusing Berlin of infringing freedom of expression.
Although officially aimed at celebrating the defeat of the military coup, the protest was also viewed by many in Germany as a sign of the leverage Mr Erdogan enjoys as a result of his support among Germany's large Turkish-origin diaspora.
The spat also comes as Germans are torn between fear for the survival of the EU's migrant deal with Turkey and dismay at Mr Erdogan's perceived post-coup crackdown on opponents.
"This is how long the reach of Erdogan's arm is," says a report in Der Spiegel on the Cologne demonstration, pointing out that the president's AK Party won a bigger share of the vote among German Turks in last year's parliamentary elections than the 49% it garnered at home.
Some German Turkish-language media reflect the mood of support for Mr Erdogan the coup has generated among many both at home and in Germany.
"They shook Europe," says the Post Aktuel newspaper of the Cologne demonstration, adding that thousands had expressed their anger at the coup, "despite the best efforts of German politicians to create obstacles".
Its tone mirrors that of pro-government newspapers in Turkey, such as Aksam, where a commentary urges Turks to "put aside their own slogans for a while and try to ensure that the language of unity dominates".
But there is dissent too. Turkey's secularist Sozcu tabloid accuses the government of having "suspended democracy" and leading Turkey into "honourable loneliness" on the international stage.
Some German Turkish media also share the misgivings about Mr Erdogan's response to the coup.
The Yeni Ozgur Politika website describes the organisers of the Cologne demonstration, the pro-Erdogan Union of European Turkish Democrats, as "fascists", and reports extensively on three counter-protests, including one by German Kurds.
A commentary by Sueddeutsche Zeitung's Turkey correspondent, Mike Szymanski, thinks the new "mistrust" between Germany is down to "one man" - Mr Erdogan, whom it accuses of reshaping Turkey into a "one-man country".
Mr Erdogan also appears to know that it is "pictures of full migrant boats in the Aegean" that will really "clobber" Angela Merkel in her campaign for re-election next year, he adds.
The jittery mood has been fanned by a demand by Turkey's Foreign Minister Mehmet Cavusoglu that the EU lift visa requirements for Turkish nationals.
This is Ankara's key condition for sticking to the migrant deal, but one the EU insists cannot be fulfilled unless Turkey modifies its terrorism and other laws.
"A compromise appears almost impossible," says tabloid Bild's website in its top story on what will happen if Turkey follows through on its threat to cancel the refugee deal. "Germany is faced with a dramatic decision."
Many feel there is no need for Germany to buckle under the pressure.
Sueddeutsche Zeitung's politics editor, Heribert Prantl, points out that the German constitutional court was right to deny Mr Erdogan the opportunity to address the Cologne rally, saying the constitution's guarantee of basic rights such as free expression only applies to citizens and not to foreign heads of state.
Die Welt's economics editor, Dorothea Sims, believes that despite Ankara's "menacing gestures", the EU money promised to Turkey in return for the migrant deal means Turkey has "too much to lose" in the case of a complete breach in relations, even if the EU never grants visa freedom.
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The row over a rally in support of Turkish President Recep Tayyip Erdogan in Germany has brought the growing tension between the countries since July's failed military coup to the surface.