Groups of settlers, who have vowed to ignore the curbs, gathered at the entrance to one settlement and said they had forced inspectors to leave.
A government official said there had been some "low level friction".
The Palestinians say Israel's 10-month building pause is not enough and are refusing to restart peace talks.
The building restrictions do not apply to East Jerusalem, where the Palestinians want to locate the capital of their future state.
'Without violence'
Settler groups have reacted angrily to Israeli Prime Minister Benjamin Netanyahu's announcement of the new policy last week.
He had been under heavy pressure from the US, with the settlements issue becoming a major sticking point in attempts to resume peace negotiations.
On Monday, the Yesha Council settler group dubbed the new policy, "illegitimate, immoral, anti-Zionist and inhuman", and said the settlers would "continue building the country with the government or without it".
On Tuesday it said it had called on all residents of Israeli settlements in the West Bank to try to "prevent, without violence"
the entry of the inspectors. An Israeli government official said that teams from the civil administration in the West Bank, which is tasked with enforcing the restrictions, had inspected 80 settlements in the past two days.
'Resistance'
He said the teams, which are operating with police escorts, had served at least 60 notices demanding that construction work be halted, and seized five heavy construction vehicles.
The official said residents and local council leaders had "showed some resistance" but "most cases were resolved peacefully".
In the settlement of Kiryat Arba in the southern West Bank, the head of the regional council, Malachi Levinger, told the Israeli media residents had forced a team of inspectors to "retrace their steps" by using "passive resistance".
"We would like them to disappear to where they came from," Mr Levinger said.
Under the Israeli new policy, backed by the security cabinet on Wednesday, permits for new homes in the West Bank will not be approved for 10 months.
In an effort to ease the fears of the settlers, many of whom are political allies of his right-wing Likud party, Netanyahu told an audience in Tel Aviv the moratorium was "a one-time decision and it is temporary".
"We shall resume building once the moratorium is over," and the future of the settlements in occupied land "shall be determined only through peace negotiations and not a single day beforehand", Netanyahu said.
Israeli courts
But municipal buildings and about 3,000 homes already under construction will still be allowed to go ahead.
Last week the Defence Ministry approved the construction of 28 educational establishments.
Separately on Tuesday, scuffles broke out at a disputed house in East Jerusalem, which a Jewish family has been attempting to take over.
Television footage showed a Palestinian hitting one of the settlers on the head with a stick. The house is one of a group of properties which both Palestinian and Jewish families claim to own. Israeli courts have recently ruled in favour of the Jewish claims in some of the cases.
UN Secretary General Ban Ki-moon expressed his "dismay" at the "continuation of demolitions, evictions and the instalment of Israeli settlers in Palestinian neighbourhoods in occupied East Jerusalem".
Nearly 500,000 Jews live in more than 100 settlements built on occupied territory in the West Bank and East Jerusalem.
Settlement building in the West Bank, including East Jerusalem, is illegal under international law - although Israel disputes this.
The Palestinians have refused to return to peace negotiations unless Israel completely ends all settlement activity.
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