NABLUS, West Bank, Feb 7 (Reuters) - International Middle East envoy Tony Blair said on Thursday the Palestinians were meeting their security obligations under a long-stalled "road map" peace plan and that Israel should start responding. The U.S. government will assess and judge whether Israel and the Palestinians are meeting their obligations under the 2003 road map as part of a push for a Palestinian statehood agreement before President George W. Bush leaves office next January. Statehood could hinge on those assessments because Israeli Prime Minister Ehud Olmert has vowed not to implement any future peace agreements until Palestinian President Mahmoud Abbas meets his road map obligation to crack down on militants. Israel has yet to fulfil its own road map commitments to halt Jewish settlement activity and to uproot outposts built without government permission in the occupied West Bank. "I think it is important to recognise that what has happened here in Nablus over these past few months is, of course, precisely what phase one of the 'road map' asks for," Blair said during a visit to the northern West Bank city. Speaking later in the West Bank city of Ramallah, Blair said "the Palestinian side have improved significantly their security capability" and cited "the need, obviously, for the restrictions on access and movement to be eased". It is unclear what direct influence Blair, who serves as special envoy to the Quartet of Middle East mediators, will have on the road map judging process, though he is close to Bush. Abbas, whose authority is limited to the occupied West Bank since Hamas Islamists seized the Gaza Strip last June, started a security campaign late last year in Nablus, seizing weapons and arresting criminals as well as some militants. Blair said progress was being made on the security front and he called for "a response from Israel with respect to its obligations" under the road map. Beyond removing outposts, Blair said Israel "in time" should remove checkpoints and other restrictions on Palestinian travel and trade. "The single most important thing for the economy is, bit by bit, to lift the occupation," he said. But, he added: "For all sorts of reasons, we know this will take time." Olmert has ruled out relaxing Israel's security grip on the West Bank until the Palestinians rein in militants, a demand likely to harden following a suicide bombing this week in southern Israel. Hamas's armed wing claimed responsibility for the bombing, which killed an Israeli woman. The group said the suicide bomber and a second attacker, killed at the scene by police, came from the southern West Bank city of Hebron. During Blair's visit to Nablus, local Governor Jamal Muheisen asked the former British prime minister for help raising $134 million for several local infrastructure projects. (Reporting by Adam Entous in Jerusalem and Haitham Tamimi in Hebron; Writing by Adam Entous; Editing by Tim Pearce)