It was no secret inside the West Wing that Bill Daley, a Catholic with deep connections to the church hierarchy, vehemently opposed the administration’s proposal to require church-run hospitals and universities to give their employees free contraception. But it was the way he pushed his case that aggravated some women on President Barack Obama’s senior staff, according to current and former administration officials. In early November, without consulting them, Daley set up a four-man Oval Office meeting for himself, Obama, New York Archbishop Timothy Dolan and Vice President Joe Biden, who both shared the view that the policy would sink...