The party's leader Hristjan Mickoski accused the government of securing parliamentary approval by "blackmailing" lawmakers and urged Zaev to call an early election this spring.
However, the ball is now in Athens' court as the name change will be effective only once it is cleared by the Greek parliament.
Eighty-one deputies in the 120-seat parliament voted in favour of the change.
Greece has previously blocked its neighbour's accession to North Atlantic Treaty Organisation and the European Union ever since Macedonia broke away from the former Yugoslavia in 1991.
Prime Minister Zoran Zaev said the move would "open the doors to the future, Macedonia's European future".
Prime Minister Alexis Tsipras' office said he spoke to Zaev after Friday's vote.
Macedonia's parliament passed an amendment to the constitution on Friday to rename the country Republic of North Macedonia in line with an agreement with Greece to put an end to a 27-year-old dispute.
Macedonian approval of the name change does not end the issue though.
Representatives of the conservative opposition VMRO-DPMNE boycotted the vote.
Tsipras has said once Greece is formally notified by Macedonia of the constitutional changes, he will swiftly launch the process of ratifying the agreement in Greece's parliament. "The vote on constitutional amendments that changed the name, identity, history and culture was done against the constitution".
The junior coalition partner in Greece's government opposes the deal, but Tsipras has voiced confidence he will be able to secure ratification with the backing of opposition lawmakers.
Greece is bound by terms of the deal to stop blocking Macedonia from NATOand other worldwide groups and to allow it to start European Union accession talks as part of the deal.