She urged Facebook to "ensure that such hate, such dirt no longer appears on Facebook pages".
A migrant surge is challenging Germany.
The authorities expect as many as 800,000 migrants - most of them refugees from Syria, Iraq and Afghanistan - to arrive this year.
`Dirt` from trolls
The Greens` leader pointed her Facebook readers to the Greens` anti-racism campaign on Twitter, called #NoHateSpeech.
"Your dirt spurs me on," she warned those who had insulted her on Facebook.
She said she was also speaking out on behalf of the many Germans helping refugees, who could not defend themselves openly.
Among the personal insults she read out, which she described as "dirt", were: "thick as a sow, you old bag", "Yank-loving slut" and "you fat dopey rat, you lack a..."
Germany is taking in far more refugees than any other EU country, as the ruling conservative-social democrat coalition says it is a humanitarian duty. The Greens support that policy.
Merkel scolds Facebook
In its "Community Standards" section, Facebook says it "removes hate speech, which includes content that directly attacks people based on their race, ethnicity, national origin, religious affiliation, sexual orientation, sex, gender or gender identity, or serious disabilities or diseases".
But Facebook goes on to say that "as with all of our standards, we rely on our community to report this content to us".
German Chancellor Angela Merkel also called for better Facebook enforcement of the rules against hate speech.
German Interior Minister Heiko Maas is expected to discuss the matter with Facebook executives on Monday.
"The rules need to be implemented," Mrs Merkel told the Rheinische Post newspaper.
"When people incite others on social networks, using their real name, not only the state must act, but also Facebook as a company must do something against those slogans," she said.
Wider threats
On Twitter, a German Green MP of Turkish origin, Ozcan Mutlu, posted a racist hate message he had received via email, which said "it`s time to get Auschwitz, Buchenwald, etc working again - that`s where you filthy Turks belong!"
He said he had asked state prosecutors to take action over it, but they had rejected his request.
Separately, a German district council chairman, Erich Pipa, said he had received a death threat from neo-Nazis, linked to his activism on behalf of migrants.
Mr Pipa, the Social Democrat (SPD) leader in the central German Main-Kinzig district, said Sunday was the day he would be targeted, according to the threat.
But he vowed to attend a local sports event as planned, regardless, the Frankfurter Rundschau reported.
Some other German politicians who actively support the integration of migrants have also been threatened. A car belonging to one - Michael Richter - was blown up in Saxony, eastern Germany, recently.
And Markus Nierth, the mayor of Troeglitz in Saxony-Anhalt, resigned after he and his family were threatened by far-right extremists.
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