In their petitions, the residents of Ankara said the decision and other reported irregularities were in open violation of the law.
In an address to legislators from Erdogan's ruling Justice and Development Party (AKP) earlier on April 18, Prime Minister Binali Yildirim said that "the people's will has been reflected at the ballot box, and the debate is over".
Worldwide monitors from the Organization for Security and Cooperation in Europe said the move undermined safeguards against fraud.
"Such a speculative statement from a spokesperson can not be accepted", Turkey's EU Affairs Minister Omer Celik told reporters at a televised press conference, calling on the EU to "respect democratic processes".
Germany is Turkey's most important trading partner, with a total volume of 37 billion euros ($39 billion), and one of its largest sources of foreign investment.
Omer Celik defended what he said were Turkey's strong legal framework and transparent process for elections.
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.
In Ankara, hundreds of people were queuing outside the election board's offices to submit petitions requesting the board reverse its decision to accept the ballots without official stamps.
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CHP vice chairman Erdal Aksunger said the party expects the board "to eliminate fraud and cancel the referendum".
Earlier Monday, the White House had sidestepped questions on the referendum, with spokesman Sean Spicer saying the US wanted to let an global commission review the results. "This is the reddest of all red lines".
President Muhammadu Buhari has congratulated the people and government of Turkey on the successful conclusion of the country's referendum on Monday.
Celik slammed the report by the global observers as "groundless" and "far from maturity".
Turkey's main opposition party has formally appealed for the annulment of a referendum to expand President Recep Tayyip Erdogan's powers, defying the prime minister's call for political forces to respect the official results of the disputed vote.
The spokesperson for the European Commission, Margaritis Schinas, said the call comes after Turkey rejected worldwide observers' criticism of the referendum on Sunday, in which over 51 per cent of voters supported granting greater powers to President Recep Erdogan.
The U.S. State Department said it had taken note of the European monitors' concerns and looked forward to a final report, urging the Turkish government to protect the rights and freedoms of all citizens, however they voted.
Mr. Buhari congratulated Turkish President, Recep Tayyip Erdogan, for the "foresight and maturity of leading his people to the polls to decide on the future of leadership for the country that will further deepen peace and stability".
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