In reaction, Greece's Defence Minister Panos Kammenos resigned early January 13 and a vote of confidence in the remaining SYRIZA government is expected to be announced shortly.
Prime Minister Alexis Tsipras continues to declare his belief that the deal will be ratified with the support of a few centre-left and independent lawmakers, filling in for the loss of ANEL support.
Kammenos, who is head of the nationalist Independent Greeks party (ANEL) which has seven MPs, said he was "sacrificing" his post over the Macedonia name issue.
Greek Prime Minister Alexis Tspiras on Sunday called for an "immediate" confidence vote in parliament over the resignation of a top minister, after his coalition government collapsed in a row over a name change deal with Macedonia.
The two countries struck the deal on the new name in June past year, but Macedonia will start using it only after the parliament in Athens also approves the change.
New Democracy is vehemently against the agreement but Tsipras hopes to convince opposition MPs from smaller parties to back the historic name change deal.
Responding to the resignation, Tsipras said that he will ask for a vote of confidence in Parliament in the coming week. A Greek government spokeswoman said the timeline for possible North Atlantic Treaty Organisation ratification will be clarified next week.
Talking to reporters after their meeting, Tsipras thanked Kammenos for his "irreplacable and significant" contribution to the coalition government, and said that their partnership was frank and fruitful. It now needs backing from the Greek parliament to come into effect.
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"Our parliament found the strength but it wasn't easy. But I am convinced that the Greek parliament will also find the strength to make the decision", Zaev told a press conference in Skopje.
The disagreement, which came to the fore after the smaller country gained its independence from Yugoslavia in 1991, has seen Athens block multiple attempts by Skopje to join North Atlantic Treaty Organisation and the European Union.
Zaev, who came to power in May 2017, is now looking to his Greek counterpart Alexis Tsipras to uphold his end of the deal, which the pair brokered past year.
"Within 10 days... if we see everything is in order, we will vote", Tsipras said on Friday evening.
Athens has for decades objected to the Former Yugoslav Republic of Macedonia's (FYROM) use of the word "Macedonia", arguing it suggests Skopje has territorial claims on the Greek border region of Macedonia.
The accord aims to start unraveling one of the world's longest diplomatic disputes.
For many Greeks, Macedonia is the name of a history-rich northern province that was the cradle of Alexander the Great's ancient empire. It began almost three decades ago, with Macedonia's declaration of independence, but has roots dating back centuries. Their efforts brought the pair a nomination for the 2019 Nobel Peace Prize.
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