The "no" camp was led by main opposition CHP and supported by the HDP.
The U.S. response has been different, with President Donald Trump calling Erdogan shortly after the referendum to congratulate him on his win.
Opposition parties in Turkey have called foul on the referendum, demanding an annulment.
The demonstrations began in smaller numbers after Sunday's vote with clattering pots and pans, but have since drawn thousands of participants expressing outrage at alleged voting irregularities.
(AP Photo/Petros Karadjias). People walk in central Istanbul's Taksim Square, Tuesday, April 18, 2017.
Monitors from the Organization for Security and Cooperation in Europe, who also listed numerous irregularities, said the move undermined important election safeguards, drawing a harsh rebuke from Erdogan.
"Rumours of irregularities in the referendum are a vain effort to cast doubt on the result".
"The people's will has been reflected at the ballot box, and the debate is over", Mr Yildirim told his parliamentary group.
Opposition parties have promised to challenge the outcome. "It is wrong to speak after the people have spoken".
In addition, changes to Turkey's executive branches will effectively abolish the position of prime minister, turning Erdogan's presidential role from a largely ceremonial one to an active one.
The new system takes effect at the next election, now slated for November 2019. However, numerous changes are expected to be implemented much sooner.
"Erdogan, whose narrow victory laid bare the nation's divisions, told flag-waving supporters that foreign election observers should "know their place" and Turkey did not "see, hear or acknowledge" criticism that the vote did not live up to worldwide standards", reported Reuters.
"The crusader mentality in the West and its servants at home have attacked us", he said.
Meanwhile, OSCE monitors were seen entering the Supreme Electoral Board headquarters. At the height of the migrant crisis, the European Union struck a deal with Turkey to provide visa-free access to the Shenzhen area and several multi-billion euros in order to assist Ankara with refugee crisis and Turkey, in turn, promised to take back refugees.
The European Union's offer of a revamped, broader trade pact with Ankara could still act as leverage to slow Erdogan's ambitions, however, a second Western official said. "We share our report and we completed our mandate".