After a nearly 30-minute meeting with Tsipras at the Prime Ministry in Athens, Kammenos told reporters they took part in the government of national unity and pulled Greece from memorandums, but cannot resume their cooperation regarding the deal.
Kammenos has long opposed the accord with neighboring Macedonia signed by their governments last year.
"The Macedonia name issue... doesn't allow me not to sacrifice the minister's chair," Kammenos said. He said he would also pull six other ministers from his party out of the government.
The impact of the resignation on the governing coalition was not immediately clear. Parliamentary elections are due in Greece by October. Tsipras's coalition has 153 seats in the 300-strong parliament, 145 of them from his leftist SYRIZA party.
Tsipras hopes to rely on lawmakers from the small pro-EU To Potami (The River) party to get the agreement approved, while some of ANEL's lawmakers also remain ambivalent to the deal.
Kammenos, who forged a coalition pact with Tsipras in 2015, has never hidden his hostility to the deal with Skopje, which renames the tiny Balkan country Republic of North Macedonia. Greece has a province called Macedonia and long demanded Skopje change its country name to remove what Athens considered to be an implied claim to Greek sovereign territory.
Macedonian lawmakers voted to rename their country the Republic of North Macedonia on Friday and the agreement now needs backing from the Greek parliament to come into effect.
Macedonian Prime Minister Zoran Zaev on Saturday called on Greece to ratify the deal.
"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," he said.
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.
A New Democracy member allegedly revealed the personal mobile phone numbers of ANEL lawmakers. Athens News Agency said the news reports prompted a Greek prosecutor to order a preliminary investigation into the allegations.
The investigation, according to judicial sources, is looking into whether personal data was violated as part of the probe.