Under the insult law, tens of thousands have been prosecuted for targeting Erdogan since he became president in 2014 after more than a decade as prime minister.
Erdogan told a group of journalists in Qatar that better ties with Israel would be 'beneficial' for peace in the wider region.
Erdogan named Nureddin Nebati, who was deputy minister, to the post, according to an announcement published in the Official Gazette.
The alliance has been tested recently by a threat to no longer recognise the US envoy and Turkey's purchase of the Russian-made S-400 missile defense system.
Describing the statement as an "impudence," Erdogan said he had ordered the ambassadors be declared undesirable.
The leaders discussed Turkey’s relations with Germany and the European Union, as well as Syria, Libya and the eastern Mediterranean.
Erdogan said he and Biden had failed to bridge their differences at a meeting during his visit to New York for the UN General Assembly this week.
The Turkish President, a close ally of Pakistan, had repeatedly raised the issue of Kashmir in his address to the high-level General Debate.
The two discussed their countries' relations and regional issues, according to a statement from the Turkish presidency. The statement did not say when the conversation took place.
A cultural center at Taksim, which was named after Ataturk and seen as a symbol of his heritage, has been demolished and is being replaced by a new structure.
Erdogan also renewed a call for the international community to take concrete steps to show Israel the "dissuasive reaction and lesson it deserves."
Turkish President picture puts Mesut Ozil under pressure for Germany's FIFA World Cup opener against Mexico
Ozil and Manchester City's Ilkay Gundogan were whistled by German fans in pre-World Cup friendlies after posing for a photo alongside President Recep Tayyip Erdogan last month.
Erdogan said he is extremely worried that some countries confident of their military power are turning Syria into a scene for arm wrestling.
Erdogan and Russian counterpart Putin agreed to hold the summit in Istanbul alongside Iran's Hassan Rouhani in telephone talks Thursday.
President Recep Tayyip Erdogan vowed Turkish troops would expand a cross-border military operation to the key strategic hub.