"Know your place," he said, after the observers said the president had been favoured by an "unequal" campaign.
The narrow victory was ruled valid by Turkey's electoral body, despite claims of irregularities by the opposition.
US President Donald Trump congratulated Mr Erdogan on his victory, as 51.4% of voters supported the changes.
In a phone call, Mr Trump also thanked the Turkish leader for supporting the US missile strike on a Syrian government airbase on 7 April, the White House said.
In a separate development, Turkey extended the state of emergency for three months. The measure, introduced after a failed coup last July, was set to expire in two days.
Monitors: 'Unlevel playing field'
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 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
"In general, the referendum did not live up to Council of Europe standards," said Cezar Florin Preda, head of the Council of Europe delegation.
The council is a pan-European human rights body of which Turkey is a member.
The monitors also criticised a late change by electoral officials that allowed voting papers without official stamps to be counted. They said this move "removed an important safeguard and were contested by the opposition.
But the head of Turkey's electoral body, Sadi Guven, said the unstamped ballot papers had been produced by the High Electoral Board and were valid. He said a similar procedure had been used in past elections.
Turkey reacts: 'Politically motivated reports'
Addressing supporters in the presidential palace in Ankara, Mr Erdogan said 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.
He 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.
Earlier, Turkey's foreign ministry said the remarks by the monitors lacked impartiality. "Saying the referendum fell below international standards is unacceptable," it said in a statement.
The main opposition Republican People's Party (CHP) has demanded a recount of 60% of the votes. Its deputy head said the result should be annulled altogether.
The pro-Kurdish Peoples' Democratic Party (HDP) also challenged the vote.
What now? By Mark Lowen, BBC News, Ankara
This was a damning verdict by the observer mission.
From biased media coverage to a misuse of administrative resources by the president and government; and from restrictions on free speech to a decision to accept unstamped ballots, which "contradicted the law"; the election observers concluded that the referendum did not fulfil international standards. So what now?
It's sure to embolden the opposition in its attempts to challenge the results. But that is an uphill struggle in a country where the state machinery is so heavily controlled by the president and his inner circle and where around 80% of the media is pro-government, pushing its interpretation of the facts.
What's more certain is that it'll shape the response to the referendum by European leaders and officials. They won't rush to congratulate a victory whose legitimacy has been placed in serious doubt. And as government supporters double down, convinced of their win, the chasm between Turkey and the west shows no sign of narrowing.
Three largest cities say No
Turnout for the divisive vote was high - 85%. Mr Erdogan won in his central Anatolian heartland and on the Black Sea coast.
But he lost in the districts containing Turkey's three largest cities.
In Istanbul, the largest city, and the capital, Ankara, the vote was very close. But in Izmir, the third-largest city, the margin was a much higher 68.8% No.
Along the Aegean and Southeast Anatolia coast - which is home to many Kurds - most districts also rejected the changes.
The constitutional changes
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 Pennsylvania-based preacher he blames for the failed coup in July
The president will decide whether or not impose a state of emergency