WASHINGTON — Under fierce election-year fire, President Barack Obama on Friday abruptly abandoned his stand that religious organizations must pay for birth control for workers, scrambling to end a furor raging from the Catholic Church to Congress to his re-election foes. He demanded that insurance companies step in to provide the coverage instead.Obama’s compromise means ultimately that women would still get birth control without having to pay for it, no matter where they work. The president insisted he had stuck by that driving principle even in switching his approach, and the White House vehemently rejected any characterization that Obama had retreated under pressure.Yet there was no doubt that Obama had found himself in an untenable position. He needed to walk back fast and find another route to his goal.The controversy over contraception and religious liberty was overshadowing his agenda, threatening to alienate key voters and giving ammunition to the Republicans running for his job. It was a mess that knocked the White House off its message and vision for a second term.Leaders from opposite sides of the divisive debate said they supported the outcome — or at least suggested they probably could live with it. Cardinal-designate Timothy Dolan of New York, the head of the nation’s Roman Catholic bishops and a fierce critic of the original rule covering hospitals and other employers, said the bishops were reserving judgment but that Obama’s move was a good first step.