Maier claims he did not write the tweet, which has since been deleted, saying one of his staff had posted it.
In a statement issued Monday, Maier apologised to Becker for the "lapse" by his employee, who he said no longer worked for him.
"This tweet not only contradicts my style but also does not reflect my ideas. I will ensure that this does not happen again," he said.
Maier's explanation appeared to have been accepted by the party's leadership, which warned him to take greater care in managing his employees.
It was a second time in a week that AfD deputies had come under fire after police filed a complaint against senior party member Beatrix von Storch over a New Year's Eve tweet which they say violated laws against incitement to hate.
Von Storch had criticised Cologne police for sending a New Year's greeting in Arabic on Twitter, asking if authorities had meant "to placate the barbaric, Muslim, gang- raping hordes of men?"
On January 1, a law against online hate speech went into effect in Germany, requiring social media companies to remove illegal inflammatory comments or face up to 50 million euros ($60 million) in fines.
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