In addition to discussing Erdoğan's win, the two leaders spoke about the recent US strike against a Syrian air base, which Trump thanked Erdoğan for supporting, and the continuing fight against the Islamic State (ISIS).
Observers from the Organization for Security and Cooperation in Europe (OSCE) and the Parliamentary Assembly of the Council of Europe (PACE) said the legal framework for the referendum "remained inadequate for the holding of a genuinely democratic referendum".
Returning in triumph to his presidential palace in Ankara, Erdogan angrily rejected the criticism, telling the monitors: "Know your place".
In a joint statement with German Foreign Minister Sigmar Gabriel, Merkel said: "There need to be talks about these issues as soon as possible with Turkey on a bilateral basis, as well as between Turkey and European institutions".
"Because we do not have the authority to forgive the murders of our martyrs", Erdogan said at a rally Monday.
Election authorities have said preliminary results showed 51.4 percent of voters had backed the biggest overhaul of Turkish politics since the founding of the modern republic, a far narrower margin than Erdogan had been seeking.
Angry opposition groups have cried foul and demanded a recount.
The move comes despite protests from opposition parties and worldwide monitoring groups about voting irregularities during Sunday's referendum.
Bulent Tezcan, deputy chairman of the Republican People's Party, announced the move at the Ankara offices of the electoral board.More news: Oklahoma deputy dies after shooting, suspect arrested
There were protests in Istanbul with a few thousand people crowding the anti-Erdogan Besiktas and Kadikoy districts, blowing whistles and chanting "We are shoulder to shoulder against fascism".
Media captionProtesters march through the Turkish capital calling for a recount Why are worldwide monitors concerned?
"With this illegal decision, ballot box councils (officials at polling stations) were misled into believing that the use of unstamped ballots was appropriate", the Union of Turkish Bar Associations (TBB) said in a statement.
"Turkey for the first time in its history has decided with the will of the parliament and its people on such an important change".
But Erdogan said Turkey had no intention of paying any attention to the monitors' report.
There were also signs of a looming crisis with the EU. Turkey's minister for European Union affairs lashed out at the EU's "unacceptable" appeal for a probe into alleged irregularities in the referendum on boosting President Recep Tayyip Erdogan's powers.
The amendments to the constitution proposed at the referendum will allow the Turkish president to stay as the head of the political party he or she represents, which is not permitted under current legislation.
The reforms will hand executive powers to the president and abolish the post of prime minister.
The referendum campaign was heavily weighted in favour of the Yes campaign, with Mr Erdogan drawing on the full powers of the state and government to dominate the airwaves and billboards.