In television images widely broadcast in Turkey, Celikkol was seen seated on a low couch, accentuating the sense of a dressing-down.
"We expect an explanation and apologies from Israeli authorities for the attitude against our Tel Aviv ambassador ... and the way this attitude was reflected," the Turkish foreign ministry said in a statement.
"We call on the Israeli foreign ministry, whose behaviour and attitude towards our Tel Aviv Ambassador did not comply with diplomacy, to obey courtesy rules."
Turkey, as a Muslim state, had been an important ally of Israel. But relations have soured following strong criticism by Recep Tayyip Erdogan, the Turkish prime minister, of Israel's war in the Gaza Strip last year.
"My protest of the attacks against Israel in Turkey still stands," Ayalon said on Tuesday.
"However, it is not my way to insult foreign ambassadors and in the future I will clarify my position by more acceptable diplomatic means."
Erdogan angered Israel this week when he accused it of being a threat to world peace.
A Turkish television drama, The Valley of the Wolves, which depicts Israeli security forces as kidnapping children and shooting old men, has added to the hostilities.
That followed a drama aired on Turkish state television last October that portrayed Israeli soldiers shooting a Palestinian baby at close range.
Israel has said the programmes are anti-Semitic and inflammatory.
Ayalon had summoned Celikkol on Monday to express Israel's displeasure over The Valley of the Wolves.
At the beginning of the conversation with the Turkish envoy, Ayalon told cameramen in Hebrew: "Pay attention that he is sitting in a lower chair ... that there is only an Israeli flag on the table and that we are not smiling."
In a subsequent interview to Israel's Army Radio, Ayalon said: "In terms of the diplomatic tactics available, this was the minimum that was warranted given the repeated provocations by political and other players in Turkey."
An official in the office of Benjamin Netanyahu, the Israeli prime minister, said he had known that Celikkol was being summoned, but that he had been unaware of the manner in which the meeting would be held.