The main opposition party rejected the result and called for the vote to be annulled.
As reported by the Reuters news agency, Erdogan repeated his intention to review Turkey's suspension of the death penalty, a step which would nearly certainly spell the end of Ankara's European Union accession process.
The statement did not say whether Mr Trump had raised independent reports of voting irregularities during the Turkish referendum or the government's heavy-handed tactics in the weeks leading up to it, when the country was under a state of emergency.
Anti-government demonstrators shout slogans during a protest in the Kadikoy district of Istanbul, Turkey, April 17, 2017.
Cezar Florin Preda, head of a delegation from the Council of Europe, said it "did not live up to. standards" because of an inadequate legal framework, a "skewed pre-vote campaign" in favour of Yes and intimidation of the opposition.
This provoked a warning from the office of the French president that any referendum on restoring the death penalty would "obviously mark a break with the values" Turkey had accepted in joining the Council of Europe.
The OSCE's head of mission, Tana de Zulueta, said on Monday that freedom of expression was curbed during the campaign, that the conditions of the vote fell "well short" of worldwide standards, and that some election monitors were inhibited from doing the job they were invited to do.
President Trump called and congratulated President Erdogan, who campaigned on behalf of the reforms and will surely be the chief beneficiary of the changes.
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Turkey's foreign ministry dismissed the observers' criticism as lacking objectivity and impartiality.
In its statement, the U.S. state department said: "We look to the government of Turkey to protect the fundamental rights and freedoms of all its citizens, regardless of their vote on April 16th, as guaranteed by the Turkish constitution and in accordance with Turkey's worldwide commitments, such as under the Helsinki Charter".
"They have a right to have elections and their people participated in that", Spicer said.
Last year, Obama said there were trends in Turkey that he was "troubled with", including a fierce crackdown on the press.
Erdogan said that "the decision is the people's".
Although a small majority of Turks voted in favour of a referendum giving greater powers to president Recep Tayyip Erdogan, in the Netherlands almost 71% voted yes, broadcaster NOS said on Monday.
Relations with Europe were strained during the referendum campaign when Germany and the Netherlands barred Turkish ministers from holding rallies. Erdogan provoked a stern German response by comparing those limits to the actions of the Nazis.
The president will be able to appoint ministers, senior government officials and to hold sway over who sits in Turkey's highest judicial body, as well as to issue decrees and declare states of emergency.