Opposition CHP head Kemal Kilicdaroglu accused the electoral board of bias and of favouring the governing party.
On Tuesday, the main opposition Republican People's Party (CHP) and the pro-Kurdish Peoples' Democratic Party (HDP) requested the poll be scrapped due to alleged violations, including vote rigging.
The plan, put forward by the ruling Justice and Development Party (AKP), gives Turkish President Recep Tayyip Erdogan sweeping and largely unchecked powers. He said the referendum result was a "clear signal against the European Union".
"The main opposition party not recognizing the results is not an acceptable thing", Yildirim said.
The CHP has led criticism of the April 16 referendum result, particularly the YSK's decision to include ballot papers unverified by local electoral boards in the count. Monitors from the Organization for Security and Cooperation in Europe, who also listed numerous problems, said the move undermined important election safeguards.
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Demonstrations have been held across Turkey to protest the YSK decision to consider the unstamped ballots valid. During that incident, parliament was bombed and Erdogan narrowly escaped assassination.
Ankara is also pressing Washington hard to extradite Fethullah Gulen, the US-based Islamic preacher it accuses of being behind last year's failed coup. The "no" campaign said it faced intimidation and threats of violence, while opposition figures and journalists were jailed.
The voters also turned Turkey's parliamentary system into a presidential one.
"I am a mortal really, I could die at any time", Erdogan said during the interview.
The referendum was seen as crucial not just for shaping Turkey's political system but also the future strategic direction of a nation that has been a North Atlantic Treaty Organisation member since 1952 and a European Union hopeful for half a century. "This small margin is unfortunate for Turkey, as such a drastic change is preferred to have a broader agreement in society", said Sinan Ulgen, chairman of think tank EDAM.
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The Turkish government said in a press release that Trump called Erdogan to congratulate him on the win and to thank him for supporting the United States' missile strike on a Syrian air base earlier this month.
Cavusoglu said that he and U.S. Secretary of State Rex Tillerson would finalize the date according to the two presidents' schedules.
Trump's phone call contrasts sharply with European leaders who have sharply criticised the referendum describing the narrow victory as a sign of authoritarianism.
The latter told CNN on Tuesday that he wanted to meet with Trump.
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