Update, July 8, 2014: This two-year-old post has been getting a lot of attention the past few days. Here is a followup on what's happened with the contraception mandate since this post was originally published.
After two solid weeks of Republicans rapidly escalating attacks on contraception access under the banner of "religious freedom," Obama finally announced what the White House is proposing: an accommodation of religiously affiliated employers who don't want to offer birth control coverage as part of their insurance plans. In those situations, the insurance companies will have to reach out directly to employees and offer contraception coverage for free, without going through the employer. Insurance companies are down with the plan, because as Matt Yglesias explained at Moneybox, contraception actually saves insurance companies money, since it's cheaper than abortion and far cheaper than childbirth. Because the insurance companies have to reach out to employees directly, there's very little danger of women not getting coverage because they are unaware they're eligible.
That's the nitty-gritty. The fun part of this is that Obama just pulled a fast one on Republicans. He drew this out for two weeks, letting Republicans work themselves into a frenzy of anti-contraception rhetoric, all thinly disguised as concern for religious liberty, and then created a compromise that addressed their purported concerns but without actually reducing women's access to contraception, which is what this has always been about. (As Dana Goldstein reported in 2010, before the religious liberty gambit was brought up, the Catholic bishops were just demanding that women be denied access and told to abstain from sex instead.) With the fig leaf of religious liberty removed, Republicans are in a bad situation. They can either drop this and slink away knowing they've been punked, or they can double down. But in order to do so, they'll have to be more blatantly anti-contraception, a politically toxic move in a country where 99% of women have used contraception.
My guess is that they'll take their knocks and go home, but a lot of the damage has already been done. Romney was provoked repeatedly to go on the record saying negative things about contraception. Sure, it was in the frame of concern about religious liberty, but as this incident fades into memory, what most people will remember is that Republicans picked a fight with Obama over contraception coverage and lost. This also gave Obama a chance to highlight this benefit and take full credit for it. Obama needs young female voters to turn out at the polls in November, and hijacking two weeks of the news cycle to send the message that he's going to get you your birth control for free is a big win for him in that department. I expect to see some ads in the fall showing Romney saying hostile things about contraception and health care reform, with the message that free birth control is going away if he's elected. It's all so perfect that I'm inclined to think this was Obama's plan all along.