Erdoğan, who has for years spoken of his desire to visit the Palestinian enclave, said last week he would go there after an official visit to the United States next month.
However, Erdoğan's plan to visit the Hamas-controlled Gaza Strip sometime in May after his trip to Washington has raised concerns. The State Department has said such a journey would be unhelpful. Both Israel and Abbas, whose Palestinian Authority is based in the West Bank, are opposed to the trip.
Kerry's comments came after he met on Sunday with Palestinian President Mahmoud Abbas in İstanbul in a bid to cement and speed up an improvement in relations between Turkey and Israel as well as explore new ways to relaunch Mideast peace efforts -- President Barack Obama's foreign policy priority for his second term.
On a trip to Israel last month, Obama secured a pledge from Turkish and Israeli leaders to normalize ties that broke down after a 2010 Israeli raid on a Gaza-bound aid flotilla that killed eight Turks and a Turkish-American. However, the rapprochement has been slow, sparking concerns that Turkey may be backsliding on its commitment.
US officials say they are hopeful that a meeting this week of Israeli and Turkish diplomats to discuss Israeli compensation for victims of the flotilla raid will jumpstart the process of restoring full diplomatic relations and exchanging ambassadors between the two countries that Washington sees as key strategic partners in the volatile Middle East.
Meeting with Kerry behind closed doors at the İstanbul-based Conrad Hotel, President Abbas made no comments to the press while leaving the hotel.
Before Abbas' visit to Turkey, news was released that Abbas wants to convince Erdoğan to cancel his Gaza visit because the Palestinian Authority is concerned that such a visit would deepen the already-existing divisions between Palestine's two rival political groups, Hamas and Fatah.
Kerry alone at a news conference said he had a "prolonged and constructive" discussion with Davutoğlu about "the importance of completing the task with respect to the renewal of relations between Turkey and Israel."
Kerry said he believes that Davutoğlu and Turkish Prime Minister Recep Tayyip Erdoğan "are deeply committed to fulfilling all the obligations of that understanding. I think they are committed to doing so."
With Abbas, Kerry was talking about ways to improve Palestinians' living conditions as a confidence-building measure to improve the atmosphere for a resumption in the stalled peace talks with Israel.
Kerry has said he fears there is only a two- or three-year window of opportunity to reach a deal on a two-state solution that would end the decades-old Israeli-Palestinian conflict and wants to move as quickly as possible. He has met with Abbas and Israeli Prime Minister Benjamin Netanyahu several times already to that end since becoming secretary of state.
Kerry was in Istanbul primarily to attend an international conference on Syria that began on Saturday afternoon and stretched into early Sunday morning as participants debated how best to boost aid to rebels trying to topple Syrian President Bashar al-Assad's regime.