One resident, speaking on Israel Army Radio, called for revenge attacks against the Israeli soldiers.
Ehud Olmert, the outgoing Israeli prime minister, called for a crackdown on the settlers involved and told his cabinet that such violence would not be accepted.
"We will not show any tolerance concerning such comments, and concerning such actions," he said at Sunday's cabinet meeting.
"We are fed up with this verbal violence."
Micky Rosenfeld, an Israeli police spokesman, said that an investigation had been opened.
The outpost bulldozed by the Israeli army had been built without government authority and was lived in by Noam Federman, a ultranationalist settler leader.
Hebron has long been a pressure point of relations between Israeli's and Palestinians in the West Bank.
Israeli human rights groups and senior military officials have said that violence against Palestinians by the some of the 300,000 settlers in the West Bank has increased in recent months.
Major General Gadi Shamni, an Israeli army commander, said last month that hundreds of settlers were now carrying out attacks.
Critics have previously said settlers have been allowed to act with impunity in the West Bank, especially with regards to violence against Palestinians.
One of the conditions of a US-sponsored peace deal between Israel and the Palestinians demands that settlement activity be frozen and dozens of the about 100 settlements on land deemed necessary for a Palestinian state are dismantled.
Settlers' resolve to keep outposts in the West Bank has strengthened since Israel removed its settlements from Gaza in 2005.
Ideological settlers believe God promised the West Bank to the Jews, although not all agree on the use of violence to attain territory.