This is, of course, a matter of a great deal of debate. I would tend to argue that it does, even though I think that it is wrong for religious organizations to deny coverage for contraception to their workers.
It is undeniably true that the Catholic Church has an official policy (regardless of what its members actually do) saying that birth control is immoral. Therefore, the Church's right to the free exercise of its beliefs must necessarily include the right to avoid paying for something that it finds morally abhorrent. It does not seem right for the government to actively force a religious organization to pay money to support a practice that it believes is immoral.
Of course, one can argue this. One can say that the workers are not working in religious jobs. One can say that the organizations take money from the federal government and must therefore abide by its rules. One can also wonder what the difference is between this and a religious organization refusing to hire women on the basis that they should stay in the home.
Therefore, I see both sides in this case. I am conflict about it, but feel that on the whole it is probably not right for the government to force these organizations to pay for contraception.