The European Commission has called on Turkey to launch a "transparent investigation" into allegations of irregularities during the referendum giving the president sweeping powers.
Turkey should "consider the next steps very carefully", an EU spokesman said.
European leaders have expressed concern that the result - 51.4% in favour of the changes - has split Turkey.
President Recep Tayyip Erdogan has rejected criticism by international election observers.
The main opposition party has launched an appeal to invalidate the result.
The changes - due to be introduced before presidential and parliamentary elections in November 2019 - will turn Turkey into a presidential republic similar to the US and France. This could enable President Erdogan to stay in power until 2029.
More on the outcome:
His narrow victory was ruled valid by Turkey's electoral body, despite claims of irregularities by the opposition.
Why are international monitors concerned?
Despite saying that the voting day was "well administered", the Organization for Security and Co-operation in Europe (OSCE) and the Council of Europe criticised the referendum campaign, saying:
- It was an "unlevel playing field" and the two sides of the campaign "did not have equal opportunities"
- It was unbalanced due to the active involvement of the president and several senior officials
- It was tarnished by a number of officials equating No supporters with terrorist sympathisers
- State resources were misused
- Under the state of emergency, essential fundamental freedoms were curtailed
- Despite some measures, the legal framework remained inadequate for a genuinely democratic referendum.
The Council of Europe - which also monitored the poll - said the vote "did not live up" its standards.
The monitors also criticised a late change by electoral officials that allowed voting papers without official stamps to be counted. But the head of Turkey's electoral body said a similar procedure had been used in past elections.
The Turkish Bar Association described the move as "illegal".
And Austrian MP Alev Korun - one of the Council of Europe observers - said "there is a suspicion that up to 2.5 million votes could have been manipulated" - which would have changed the result. She offered no evidence.
What did the president say about the result?
Mr Erdogan told supporters on Monday that Turkey did not "see, hear or acknowledge the politically motivated reports" of the monitors.
The result, he said, ended the debate on changing the constitution and creating an executive presidency, adding that the process of implementing the reforms would now begin.
His prime minister - whose post is being abolished under the changes - said "rumours of irregularities" were an effort to cast a shadow over the result.
Read more: Erdogan's rise to power
Mr Erdogan also said the country could hold a referendum on its long-stalled EU membership bid.
Additionally, Mr Erdogan said he would approve the death penalty if it was supported in a referendum or a bill was submitted to him through parliament. This would end Turkey's EU negotiations.
What have European leaders said?
German Chancellor Angela Merkel warned that the "tight referendum result shows how deeply divided Turkish society is and that means a big responsibility for the Turkish leadership and for President Erdogan personally".
In their statement on Sunday, top European Commission leaders issued a similar call.
In addition to calling for transparent investigations of complaints, Commission spokesman Margaritas Schinas also urged "all actors to show restraint" in his statement on Tuesday.
Other European leaders have expressed concern about the possibility of the return of capital punishment.
Austrian Foreign Minister Sebastian Kurz went further. He said the referendum result was a "clear signal against the European Union". The "fiction" of Turkey's bid to join the bloc must be ended, Mr Kurz said.
In contrast with European leaders, US President Donald Trump has congratulated the Turkish president, hours after the US state department mentioned concerns by international observers and urged Turkey to respect the rights of its citizens - chiming with sentiment in European capitals.
What do the constitutional changes include?
- The president will have a five-year tenure, for a maximum of two terms
- The president will be able to directly appoint top public officials, including ministers and one or several vice-presidents
- The job of prime minister will be scrapped
- The president will have power to intervene in the judiciary, which Mr Erdogan has accused of being influenced by Fethullah Gulen
- The president will decide whether or not to impose a state of emergency