I can see only one scenario:
Keep the timeline the same until after the Six Day War. For some reason (maybe Moshe Dayan becomes PM), there is less settlement of the West Bank, and nearly all settlements are either close to the Green Line or concentrated in a single large bloc deep in the West Bank. Gaza is not settled at all.
Sometime in the 2000s, Ariel Sharon (or another Israeli PM) decides to pull a unilateral evacuation of the West Bank, save for those settlements close to the border. The big bloc decides to declare independence, creating the
State of Judea. Their leadership is smart enough to make conciliatory statements, and they run a slick media campaign about how their lives will be ruined if they are forced to evacuate. The settlers are armed and make it clear they will resist the evacuation with force if necessary. Sharon decides to ignore them.
The trouble is that such a state wouldn't be stable. The Palestinians would almost certainly attack them, leading to constant bloodshed. Any Israeli government that wasn't hard-right would be reluctant to support it - a left-wing government might regard such a state as a threat. My guess is the State of Judea would eventually suffer economic collapse, and most of its inhabitants would return to Israel, leaving a few diehards. If it was created after 2010, maybe it would still be around today, but it wouldn't survive much longer.