It was the first time Ankara has explicitly threatened to cut ties with Israel, having previously said it was reviewing relations with the Jewish state.
The public exchange between the two once close US regional allies followed talks last week by Turkish and Israeli officials aimed at mending fences. Instead, their positions appeared to have hardened.
"Israel has three paths ahead: It either apologizes, or accepts the findings from an international commission investigating the raid, or Turkey will cut off ties," Turkey's Foreign Minister Ahmet Davutoglu told Monday's edition of Hurriyet newspaper.
Once Israel's closest Muslim ally, Turkey has said several times it wants Israel to apologize over the May 31 raid, pay compensation, agree to a UN. inquiry into the incident and lift the blockade of 1.6 million Palestinians living in Gaza Strip.
Israeli Foreign Minister Avigdor Lieberman said his country has no intention of apologizing. "We don't have any intention to apologize. We think that the opposite is true," he told reporters during a visit to Latvia.
Davutoglu met Israel's Trade and Industry Minister Benjamin Ben-Eliezer last week in Brussels in a bid to repair ties. Turkey said then it conveyed its demands to Israel but Davutoglu told Hurriyet: "We will not wait forever for an answer."
"It will be enough if their own commission rules that the raid was unfair and they apologize in line with the commission's verdict, but we have to see the verdict first," Davutoglu said.
Nine Turkish activists were killed when Israeli commandos stormed the Turkish-flagged ship Mavi Marmara as part of an operation to stop a relief aid flotilla headed for Israeli-blockaded Gaza.
Turkey withdrew its ambassador to Israel, cancelled joint military operations and barred Israeli military aircraft from Turkish airspace after the incident.
The United States wants Israel and Turkey, whose earlier friendship had benefited US policy in the Middle East, to patch up the dispute. President Barack Obama is due to meet Israel's Prime Minister Benjamin Netanyahu in Washington on Tuesday.
