Prime Minister and AKP leader Binali Yildirim said Erdogan can rejoin the party he founded in 2001 once official results of the plebiscite, granting him sweeping powers, are announced.
The CHP has led criticism of the result of Sunday's referendum, particularly the YSK's decision to include ballot papers unverified by the local electoral board in the count.
He said that "the path to seek rights" should be limited to the courts.
The referendum asked voters to choose "yes" or "no" on 18 constitutional amendments, one that will see the country switch from a parliamentary to a presidential system.
The pro-Kurdish opposition Peoples' Democratic Party said it presented complaints about unstamped ballots affecting 3 million voters, more than twice the margin of Erdogan's victory.
Unofficial results show a narrow win for President Recep Tayyip Erdogan's "Yes" campaign, which won 51.4% of the vote.
The statement was released hours after White House spokesman Sean Spicer declined to comment on the result of the constitutional reform referendum in Turkey until a review of the vote by an global electoral monitoring mission is finishe d.
"It is clear that the High Electoral Board is not receiving its power from the people, the law or the constitution but rather from a specific centre, a specific political authority", CHP leader Kemal Kilicdaroglu said in a speech, accusing the body of "changing the rules mid-game".
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. "We will follow closely how Turkey behaves on this".
Underscoring the complicated relationship between the U.S. and Turkey, the White House readout of Trump's call also noted the pressing issues on which the USA has tried to work with Turkey, namely fighting the Islamic State group and quelling Syria's civil war.
Declining to congratulate the Turkish president, the European Union executive's second official response since the vote instead focused on observer findings that the vote was skewed in Erdogan's favour without a proper legal framework and with late changes in ballot counting.
Celia also denounced election observers' criticism that the country's referendum fell below global standards, saying their remarks lacked objectivity and impartiality.
Foreign Minister Mevlut Cavusoglu said that Erdogan and Trump would meet in person next month, before a North Atlantic Treaty Organisation summit.
European officials have firmly declined to congratulate Erdogan's win, unlike the US president.