Binyamin Netanyahu's stunning political coup this week, calling off elections and forming a unity government, was partly a response to increasingly strident demands of rightwing settlers in the West Bank coming from within his own party.

The settlers have become a vociferous presence in Israel's dominant party, Likud, as they seek to drive government and party policy into line with their beliefs. Their growing presence in the party helped push Netanyahu to scrap elections called for September and bring the centrist Kadima party into a broader coalition. Israel's parliament is poised to approve the new government.

Activists from West Bank settlements packed a Likud convention on Sunday, demanding a secret ballot to choose its chairman and denying Netanyahu control over proceedings. According to the columnist Nahum Barnea, writing in Yedioth Ahronoth, the group staged a "mutiny" and Netanyahu was "astounded by the number of kippa wearers and their domination of the atmosphere".

"The disruption by the rightwing lunatic fringe at the central event of the ruling party made the prime minister realise what he should have understood long ago: Likud has lost its mind," wrote Ari Shavit in Haaretz.

The vote was postponed, but a reportedly rattled Netanyahu subsequently took the decision to tilt his government towards the political centre and reduce the leverage of the extreme right both inside his party and the coalition.

Within Likud, militant settlers have been gaining strength for some years, but they scored major victories earlier this year when several – including radical activists from hilltop outposts – were elected to the party's central committee, and their leader, Moshe Feiglin, won around 25% of votes in a challenge to Netanyahu for the party chairmanship. Around 7% of Likud's 130,000 registered members live in West Bank settlements.

Feiglin denies he is trying to lead a takeover of Likud. "I don't know what takeover means," he told the Guardian. "Democracy means everyone can put forward alternative ideas. I've been a member of the Likud party for many years. I'm not taking something that doesn't belong to me: Likud belongs to me, just as it does to any other member of the party."

Feiglin joined Likud 12 years ago, along with "many friends and people who think the same way as I do. We are quite a strong group today."

In an indication of the group's growing influence, government ministers and senior party figures have recently flocked to Ulpana, a West Bank outpost whose demolition the supreme court has ordered after ruling it was built on private Palestinian land, to show solidarity with the settler movement.

Feiglin, once seen as a political outcast, is now feted as a power-broker.

The policies this militant settler faction is pushing revolve around a Jewish state from the Jordan river to the Mediterranean, which means Israel annexing the West Bank and Gaza and encouraging Palestinians – or Arabs, in Feiglin's terminology – to leave.

"The Arabs in the West Bank have no nationality apart from being part of the big Arab nation. They can stay in Judea and Samaria [the Biblical term for the West Bank], with rights they won't get anywhere else in the Arab world, as long as they accept Israel's sovereignty. They will have human rights but not the right to vote. If voting is so important to them, they have 22 Arab countries to choose from. If they stay without accepting Jewish sovereignty, that means war."

His political motivation was to strengthen the Jewish state, he said. "Israel is the home of the Jewish people. It is not a state of all its citizens." The reason why the Jews came back from exile to the promised land was not to establish a democracy, but to create a homeland, he added.

Feiglin pointed out that Likud's charter "flatly rejects" the creation of a Palestinian state west of the Jordan river. "It says specifically that these territories should be part of the state of Israel. I expect Netanyahu to act according to the charter."

Gershom Gorenberg, author of The Unmaking of Israel, said Feiglin's supporters had been "very successful in acting as a bloc to push the party to the right. In terms of taking over Likud, and Feiglin becoming leader, they don't really have a chance. But it's clear they exert a strong influence, and government ministers and members of the Knesset [parliament] are eager to secure their support. Netanyahu has found he doesn't control his own party."

Feiglin plans to challenge Netanyahu again for the leadership. "The time will come to put forward my candidacy. I believe one day I will lead the state of Israel," he said.