Monitoring group the Organization for Security and Co-operation in Europe questioned the fairness of the referendum in a preliminary report released on Monday, concluding the vote "took place on an unlevel playing field and the two sides of the campaign did not have equal opportunities".
The White House said they also discussed a U.S. missile strike in Syria and the fight against the Islamic State group.
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.
"The winners on April 16 were all of our people who said No or Yes by exercising their democratic choice".
An Austrian member of the Council of Europe observer mission said up to 2.5 million votes could have been manipulated, nearly double the margin of Erdogan's victory, and that the YSK decision on unstamped ballots appeared illegal.
That drew a harsh rebuke from Erdogan and criticism from Prime Minister Binali Yildirim.
In another challenge to Turkey's European partners, Erdogan said a separate referendum might be held on putting an end to its EU candidacy.
The Republican People's Party, or CHP, called for the vote on the referendum to be annulled citing a series of irregularities, particularly an electoral board decision to accept ballots that didn't bear official stamps, as required by Turkish law.
Speaking in parliament on Tuesday, Yildirim said "rumours" of irregularities were a vain effort to cast doubt on the result.
The pro-Kurdish opposition Peoples' Democratic Party said it presented complaints about unstamped ballots affecting 3 million voters, more than twice the margin of Erdogan's victory.
Even though the White House press secretary said that Washington will not comment on the topic until the worldwide monitors who oversaw the Turkish constitutional referendum process release their final report, U.S. President Donald Trump called his Turkish counterpart in order to congratulate him on his referendum victory. "Everyone should respect the outcome, especially the main opposition".
The new system takes effect at the next election, now slated for November 2019.
"For the first time in the history of the Republic, we are changing our ruling system through civil politics". This would allow Erdogan to rejoin the governing AK Party he co-founded, or to lead it. Tana de Zulueta, head of the observer mission, told reporters that the group had paid a courtesy call and held a "cordial" meeting with board members.
Mr Erdogan told supporters on Monday that Turkey did not "see, hear or acknowledge the politically motivated reports" of the monitors. "But figures already published show that Turkish society is divided on this planned important reforms", it said, calling for "a free and honest dialogue" in Turkey among all components of political and social life.
President Donald Trump, meanwhile, ignored the concerns about voting irregularities and congratulated Erdogan on his referendum victory.
Angelino Alfano in a written statement Monday also called for a "cooling of internal tensions".