As the threat from Sunni militants in western Iraq escalated last month, Prime Minister Nuri Kamal al-Maliki secretly asked the Obama administration to consider carrying out airstrikes against extremist staging areas, according to Iraqi and American officials. But Iraq’s appeals for military assistance have so far been rebuffed by the White House, which has been reluctant to open a new chapter in a conflict that President Obama has insisted was closed when the United States withdrew the last of its forces from Iraq in 2011. A spokeswoman for the National Security Council, Bernadette Meehan, declined to comment on Mr. Maliki’s...