The head of the board said it had received many complaints that polling stations didn't have stamps and made the decision to accept the ballots after an appeal from a ruling AK Party official.
Also, the Organisation for Security and Co-operation in Europe said earlier that the Turkish referendum took place on "an unlevel playing field". The assessment drew a harsh rebuke from Erdogan and criticism from Prime Minister Binali Yildirim.
"Efforts to cast a shadow on the result of the vote by spreading rumors of fraud are futile and in vain", Yildirim said. "The people's will has been reflected at the ballot box, and the debate is over", Yildirim told his parliamentary group. "It is wrong to speak after the people have spoken". He accused the board of "changing the rules midgame".
Hundreds of people are queuing in front of Turkey's election board in capital Ankara to submit petitions requesting that the electoral authority reverse a controversial decision during Sunday's referendum to accept ballots without official stamps, as required by Turkish law.
The board met for seven hours Wednesday before rejecting the request.
"We look to the government of Turkey to protect the fundamental rights and freedoms of all its citizens - regardless of their vote on April 16 - as guaranteed by the Turkish constitution and in accordance with Turkey's global commitments", the State Department said in a statement.
The referendum allows Erdogan, who has ruled Turkey since he became prime minister in 2003 and then president in 2014, to fulfill his long-held ambition for a presidency with executive powers.
Nigeria's President Muhammadu Buhari on Tuesday felicitated with the people and government of Turkey on the successful conclusion of the country's referendum on Monday.
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The new system takes effect at the next election, now slated for November 2019.
Turkey's bar association said a last-minute decision by the YSK electoral board to allow unstamped ballots in the referendum was clearly against the law, prevented proper records being kept, and may have impacted the results.
A senior official with the Trump administration said the President did not raise any concerns about the referendum with Erdogan, describing that part of the phone call as a "simple congrats".
The head of Turkey's Supreme Electoral Commission, Sadi Güven, confirmed that "yes" had narrowly clinched the victory, based on unofficial results.
Ms. Schinas said: "We call on all actors to show restraint and on the authorities to launch transparent investigations into these alleged irregularities found by the observers". "Tayyip Erdogan may have done more good than the other big players (of Turkish politics) ... but I think of Tayyip Erdogan as just the best of the worst". "We share our report and we completed our mandate".
Some 2,000 protesters in Istanbul Wednesday evening demanded the resignation of the electoral board and chanted "Don't be silent, shout out, "no" to the presidency".
Showing no sign of pulling his punches, Erdogan said Turkey could hold further referendums on its European Union bid and re-introducing the death penalty.
Trump also spoke to Erdogan in February, reports the Guardian.