"I was told that we received the response that we wanted and expected in diplomatic terms," Erdogan told reporters at the airport after returning from a visit to Russia.
"The letter includes an expression of apology," he said.
Abdullah Gul, the Turkish president, deemed an earlier Israeli apology on Wednesday as insufficient and threatened to recall his ambassador unless a formal apology was made by the end of the day.
"Unless they make up for it by this evening, our ambassador will return on the first plane tomorrow to hold consultations," Abdullah Gul, the Turkish president, was quoted as saying by the country's NTV news channel.
After Wednesday's official apology, Netanyahu "again expressed his concern over the cooling of the ties between Israel and Turkey" and instructed officials "to find ways to prevent this trend," the statement from his office said.
The series that sparked the latest row between the two countries was a Turkish television drama, called The Valley of the Wolves, which Israel says depicts Israeli security forces as kidnapping children and shooting old men.
That followed a drama aired on Turkish state television last October that appeared to portray Israeli soldiers shooting a Palestinian baby at close range.
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, the deputy foreign minister 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."
The row came ahead of a planned visit by Ehud Barak, the Israeli defence minister, to Turkey on Sunday.