Gabriel responded saying that it would be "regrettable" if Netanyahu cancelled the meeting due to what Gabriel describes as "completely normal" encounter.
Gabriel said it would be "unthinkable" to cancel meetings if Netanyahu met government critics in Germany. But he says a cancellation of Tuesday's meeting wouldn't be a "catastrophe," and wouldn't change his relationship with Israel. He added, "You can't get a proper and comprehensive picture in any country on Earth if you only meet in government offices."
Gabriel, who is also deputy chancellor, is making his first visit to Israel and is expected to continue to the Palestinian Authority and Jordan.
According to Channel 2 news, when Netanyahu learned of his planned meetings with representatives of the leftist organizations, his office sent a message to the German delegation that this was provocative behavior.
However, in an attempt to compromise with the Germans, Netanyahu's office made it clear to Gabriel that it would not object to a junior representative meeting the organizations on Gabriel's behalf.
Gabriel, a leader of the Social Democratic Party in Germany, holds firm views on Israel regarding the conflict with the Palestinians.
In the past, he accused Israel of implementing an apartheid regime in Hebron and provoked anger in Jerusalem when he traveled to Tehran immediately after signing the nuclear agreement with Iran.
Netanyahu's ultimatum comes during a period of instability between Israel and Germany highlighted by Germany cancelling an inter-governmental meeting scheduled for next month in protest of the Regulation Law and settlement construction.
The chairman of the Zionist camp, Isaac Herzog, announced that he would meet as planned with the German foreign minister and attacked Netanyahu for his ultimatum.
"Netanyahu is running away from the court. Placing the ultimatum is a serious blow to Israel's foreign relations with the largest economy in Europe and a true friend of the Jewish people.
"Instead of running away, I call upon Netanyahu to meet with the German foreign minister and present his views and positions, without fear of organizations," said Herzog.
The incident with Gabriel comes about two months after Belgian Prime Minister Charles Michel, who visited Israel earlier this year, met with representatives of Breaking the Silence and B'Tselem a day after his meeting with Netanyahu.
As a result of the meeting, Netanyahu ordered the summoning of the Belgian ambassador to the Foreign Ministry.
The Associated Press contributed to this report.
(Translated and edited by Fred Goldberg)