"The way president Trump is approaching these matters makes us happy", he said.
The opposition has cited several problems with how the vote was conducted. Everyone and all sections - and the main opposition party in particular - must show respect.
The European Union also urged a probe into the poll fraud claims after worldwide observers voiced concerns, but both US President Donald Trump and Russian President Vladimir Putin called Erdogan to offer congratulations.
The petitions were filed by the country's Republican People's Party, Peoples' Democratic Party and the Patriotic Party. Some 51.4 percent of voters cast their ballots in favor of the amendments to the Turkish Constitution.
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.
The new system takes effect at the next election, now slated for November 2019. The president will also be allowed to keep his ties to a political party.
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Prime Minister and AKP leader Binali Yildirim said Erdogan can rejoin the party he founded in 2001 once official results - expected before the end of the month - are announced.
On Tuesday, Yildirim said Erdogan would be invited to join the party as soon as the official results are declared.
Erdogan, meanwhile, called the referendum "the most democratic election. ever seen in any Western country" and admonished the OSCE monitors to "know your place".
A referendum on amendments to the Constitution of Turkey was held in the country April. 16.
EU Commission spokesman Margaritis Schinas echoed concerns about the vote's legitimacy, asking Turkish officials to "launch transparent investigations into these alleged irregularities found by the observers".
"For years, we have worked on getting ourselves integrated with the world", Serafettin Asut, head of the chamber of commerce and industry in the Mediterranean city of Mersin, home to one of Turkey's largest global ports.
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In Istanbul, hundreds of "no" supporters demonstrated in the streets on Monday, chanting "thief, murderer, Erdogan" and banging pots and pans.
The tone of the message of the USA president contrasted with the statement issued by the State Department, which cited the views of worldwide observers on 'irregularities on voting day and unequal conditions during the campaign period'.
Underscoring the complicated relationship between the USA and Turkey, the White House readout of Trump's call also noted the pressing issues on which the US has tried to work with Turkey, namely fighting the Islamic State group and quelling Syria's civil war.
Asked during the White House daily briefing to respond to Erdogan's accumulation of power after the referendum, Press Secretary Sean Spicer echoed the State Department statement and said he would withhold reaction until a final report is published.
Bilginsoy contributed from Istanbul and Mehmet Guzel contributed from Ankara.
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