Interior Minister Thomas de Maiziere said this week he was willing to discuss the possibility of introducing a Muslim holiday in parts of the country.
Germany is home to some 4.4 million Muslims, with many coming from the nation's large ethnic Turkish community. The over one million migrants that arrived in recent years also include many Muslims.
"Where there are many Muslims why shouldn't we consider a Muslim holiday," the interior minister said at a rally in Lower Saxony ahead of Sunday's regional election in the state.
The CSU, the Bavaria-based sister party of Merkel's Christian Democrats, strongly rejected the idea.
"Germany's Christian heritage is not negotiable," Alexander Dobrindt, a senior CSU politician, told the Bild newspaper.
"For us, the introduction of Muslim holidays is out of the question," he said.
Social Democrat leader Martin Schulz meanwhile said Saturday that the idea was worth "thinking about", according to the DPA news agency.
Schulz said he was surprised that the idea had come from the interior minister who he said was usually known for having "very little imagination" in this area.
De Maiziere had previously called on immigrants to respect the German "Leitkultur", culture of reference, a term regularly used by the far-right.
Sunday's regional vote is a major test for Merkel after she won a fourth term in a national election in September but without a majority in parliament, which has forced her to embark on high-stakes coalition talks.
Latest surveys show the CDU lagging behind the Social Democratic Party (SPD) in Lower Saxony, the fourth most populous state in Germany.