President Recep Tayyip Erdogan reacted furiously after the 11 German MPs with Turkish roots last week backed a parliamentary resolution that recognised the mass killings of Armenians as a genocide.
The sensitive issue has infuriated Erdogan at a time when relations are already strained by disputes about media freedom, while the EU is banking on Turkey to stop the cross-border flow of migrants.
A group of Turkish lawyers has also filed a complaint with prosectors asking for the 11 German lawmakers to be charged with "insulting Turkishness and the Turkish state," the Hurriyet daily reported.
Schulz, a German national, also condemned "in the strongest terms" Erdogan's comments linking the lawmakers to "terrorists", in reference to the outlawed Kurdistan Workers Party (PKK).
Lawmakers and journalists' ability to work "without having to fear repression is part of the non-negotiable foundation of every democracy," he wrote.
Schulz warned that if national leaders challenge these rights, this "can be damaging to international relations in the long run".
The German parliament's president, Norbert Lammert, said that statements by Turkish leaders had prepared the ground for a torrent of "hateful threats and insults" that were mailed to the MPs.
He added that "anyone who tries to pressure individual MPs with threats must know that he is attacking the whole parliament".
Lammert also warned that "we will respond accordingly with all options available to us under the law".
