However, opposition parties said the exercise was marred by irregularities, adding that they would challenge its result.
Opposition CHP head Kemal Kilicdaroglu accused the electoral board of bias and of favoring the governing party.
Speaking to a rally of supporters in Ankara on Monday he said that some European countries had opposed him winning the referendum more than members of the Turkish opposition.
In a normal moment in American political history, a president of either party would have likely responded to Sunday's vote by either offering measured comments encouraging Erdoğan to respect the views of his political opponents or avoiding the substance of the referendum altogether and simply saying that the United States respects the will of the Turkish people. "The Supreme Election Board issued instructions late in the day that significantly changed the validity criteria, undermining an important safeguard and contradicting the law", observed Cezar Florin Preda of the monitoring group at the Ankara press conference.
And in Sunday's referendum, the YSK's overseas election branch had already rejected an appeal by a ruling AK Party official to have unstamped envelopes counted as valid.
The queues formed as the main opposition party was scheduled to formally request that the electoral authority annul the referendum over the ballots lacking the official stamp.
PRESIDENT Muhammadu Buhari has felicitated with the people and government of Turkey on the successful conclusion of the country's referendum on Monday.
The referendum - if fully implemented - would change Turkey from a parliamentary democracy into one run by a strong executive president who will absorb all the current functions of the prime minister.
Turkey's main opposition began a battle on Tuesday to annul a referendum handing President Tayyip Erdogan sweeping new powers, while the bar association and an worldwide monitor said an illegal move by electoral authorities may have swung the vote.
The emergency laws first went into effect in July of a year ago after a failed coup, which Erdogan blames on the followers of US-based cleric Fethullah Gulen.
"With this illegal decision, ballot box councils (officials at polling stations) were misled into believing that the use of unstamped ballots was appropriate", the Union of Turkish Bar Associations said in a statement, according to the Reuters news agency.
The White House said they also discussed a USA missile strike in Syria and the fight against the Islamic State group.
The U.S. State Department called on Turkey to protect basic rights and freedoms as authorities work to resolve the contested results.
Relations between Turkey and Europe's powerhouse nations have come under strain over Erdogan's comments that he would seek the restoration of the death penalty - a move that would sink Turkey's long-stalled bid to join the European Union.
Worldwide monitors said the move undermined safeguards against fraud.