This probably requires at least correction or 2, so someone please make any they have:.
Basically Israeli policy towards Gaza since Sharon disengaged has been to occupy the air and sea and blockade it and bomb it whenever rockets are fired or anything violent happens from Gaza to Israel.
Israeli policy towards the West Bank is to build settlements there and take over the land piece by piece, perhaps stopping at some point, leaving a bunch of West Bank towns/cities under the control of Fatah, with no voting rights in Israeli elections. The closest comparison I've seen is the Bantustans of South Africa, because in indigenous reserves in say Canada the residents can vote in Canadian federal elections.
The settlers in the West Bank have a pretty green light to launch attacks without Israeli state organs (police, IDF) stopping them, because the attacks and land grabs are consistent with Israeli state policy. The major political opposition to these attacks and land grabs is the Israeli center, illustrated by the figure of Gantz (in the coalition now, fairly good odds to be next PM), who has supported stopping the attacks and land grabs by settlers, though he is against giving back any of the settlements created so far.
So basically, the settlers on the ground and the right in Israel and some of the center want to keep creating new settlements/land grabs. The center mostly wants settlement creation/land grabs to be decided by the state, not the settlers on the ground, or not done anymore, but also no settlements made before today to be razed or have the Jewish population evicted from those places.
The West Bank is being taken piece by piece by settlers who are generally expansionist and believe Israel should control all lands they can make a biblical claim to, but don't want a one state solution where the Palestinians get to vote in Israeli elections and can travel the country freely. They want the West Bank to be a mix of Israel, and Apartheid style Bantustans. The Netanyahu governments back them when they can get 61/120 seats without center votes and form a government, which they had before the War, but then they added in the center because Israeli convention is to be unified during a war, and likely also because the massive protests in Israel (Bibi and the right moved to reduce the power of the Supreme Court and Constitution) which meant the government was worried people wouldn't support the war effort unless someone like Gantz was in the war cabinet as well.
The center, like the center in many countries, doesn't love the right wing solution but doesn't oppose it, and effectively has no position on a long-term peace process, since that would mean either a one state solution (which they don't support) or a 2 state solution (which they sort of support in theory, but if they won't get rid of settlements, that would either be a state of patches of unconnected land (which I don't think has ever existed?) it would have to be only the Eastern half of the West Bank).
And in this specific case:
"Everything to the right" = more settlements in the West Bank, some of which are created via private settler attacks and some of which are state created.
"Block movement back to the left" = The state getting rid of those settlements.