Panos Kammenos, whose Independent Greeks party props up Tsipras minority government, announced his resignation on Sunday morning after a meeting with Tspiras.
Greek Prime Minister Alexis Tsipras called for the confidence vote after meeting with Greek Defense Minister and coalition partner Panos Kammenos in Athens.
Tsipras and Kammenos met for almost an hour before the prime minister said he had accepted Kammenos' resignation, but added that he would call for a confidence vote in parliament to keep his government alive.
The new name will be used both internationally and bilaterally, so that even the 140 or more countries that recognise the name Macedonia will also have to adopt North Macedonia.More news: Leeds United in talks to sign Real Madrid keeper Kiko Casilla?
Greece has a parliamentary election scheduled for October and Syriza is now trailing the centre-right New Democracy party, which also opposes the deal with FYR Macedonia, by 12 points. The replacement, also approved by Kammenos, will take effect Tuesday.
Meanwhile, ahead of the vote in Athens, two news websites alleged that a prominent member of Greece's main opposition New Democracy party - which rejects the name deal - called on its members to phone lawmakers of the ANEL party and urge them to vote against the agreement.
The coalition government of SYRIZA and the Independent Greeks (ANEL) party, in power since January 2015, had been widely expected to fracture over the parliamentary ratification process for the Prespes Agreement.
The nationalist ANEL has supported the Tsipras administration with its seven lawmakers and has six ministers and junior ministers in the government. Polls indicate it will probably fail to obtain the minimum three percent of the popular vote needed to meet the Greek parliamentary election threshold. "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 controversial agreement would also allow Macedonia to join North Atlantic Treaty Organisation and the European Union.More news: Winter Weather Advisory in effect; expect heavy snow in Chicago-area counties
"Within 10 days... if we see everything is in order, we will vote", Tsipras said on Friday evening. In ancient times it was the cradle of Alexander the Great's empire, a source of intense pride for modern-day Greeks. This remained the case for centuries and so this concept - of what Macedonia is - has stuck.
The naming row between the two countries began 27 years ago when FYROM declared independence from the Socialist Federal Republic of Yugoslavia, but has roots going back to antiquity.
Greece has a province called Macedonia and has repeatedly called on Skopje to change its country's name, which Athens considers an affront to Greek sovereignty.
Last June, Zaev and Tsipras reached a landmark compromise over the name dispute.More news: Alabama transfer QB Jalen Hurts visits OU's campus