Erdogan showed no sign of conciliation, despite of the narrowness of his victory.
The Organization for Security and Cooperation in Europe (OCSE) said the Turkish referendum passed on an "unlevel playing field", and the head of OCSE said freedom of expression opposing the bill had been inhibited, while the voting procedure fell "well short" of global standards.
Mr Erdogan has said he will ignore worldwide criticism of Sunday's referendum, telling foreign observers to "know their place". Opponents accuse him of leading a drive towards dictatorship.
In an exclusive interview with CNN, Erdogan said that a constitutional reform package - backed by a narrow majority of Turkey's voters in a referendum on Sunday - was not about him.
"It's close enough that the opposition says it's going to challenge alleged irregularities", he says.
Turkey's electoral board rejected claims of voting fraud shortly after victory was declared, while the country's foreign minister described the result as the birth of a "truly new Turkey". In a statement, the foreign ministry said that the observer mission's findings that Sunday's referendum procedures "fell short" of global standards "were unacceptable".
"We have put up a fight against the powerful nations of the world", he said as he arrived at the airport from Istanbul.
Erdogan has also vowed to consider reinstating the death penalty - a move that would all but end prospects of European Union membership.
Erdogan, a populist with a background in once-banned Islamist parties, has ruled since 2003 with no real rival, while his country emerged as one of the fastest-growing industrial powers in both Europe and the Middle East.
"We look to the government of Turkey to protect the fundamental rights and freedoms of all its citizens", said the department's acting spokesman, Mark Toner.
Bulent Tezcan, deputy head of the CHP demanded the referendum be reheld, saying that would be the "only decision that will end the debate about the legitimacy" and ease people's concerns.
Although the margin fell short of the sweeping victory Mr Erdogan had sought in the landmark referendum, it could nevertheless cement his hold on power in Turkey.
Turkey has already been under a state of emergency for nine consecutive months after a coup attempt on July 15, 2016.
He made the call Monday after the "Yes" campaign won over 51 percent of the votes during Sunday's referendum, while the "No" campaign gained almost 49 percent.
Opposition parties complained of a number of irregularities in the voting, and were particularly incensed by an electoral board decision announced on Sunday afternoon to accept as valid ballots that did not bear the official stamp.
"There is talk and suspicion about up to 2.5 million ballots that could have been manipulated", she told Austrian public radio Oe1 from Istanbul.
The new system is due to come into effect after elections in November 2019.