It's the Bishops who've made a bad strategic move.
He's not "caved in", he used the issue to set one more trap for the moral right in the middle of the Republican race, which already many of them have already fallen for hook, line and sinker (notably Santorum), which was precisely Obama's real goal. He made a strategic retreat, and a clever one at that.
He wasn't even elected, we didn't even foresee the shift of power in Congress, or the depth of the upcoming economical crisis that most professional observers of American politics everywhere we're already saying that Obama would have a very rough, fairly disappointing first mandate, that the voters would have to be very very patient with him as most his ambitious promises he wouldn't be able to deliver much on unless he got his second mandate and got clear of the risk to sabotage his hopes to get one, and even then. And that he'd better prey for the economy to recuperate some and hold, or the focus would shift almost exclusively to the Democrat vs. Republican economic visions, and he might be finished. The Republicans can't really attack Obama on his failures on his promises to his voter base on other fronts than economy, they're the ones who fought to make him fail, who forced him to either totally cave in and disappoint his base, or make stands and risk that the results of his policies don't deliver enough before the next election, or that the adjustement phase of his programs frighten enough people into not giving him a second mandate to complete them, and go further (he's trying to bring about over 8 years changes that were accomplished far more gradually, and sometimes took decades to complete elsewhere). He's made tons of mistakes, yes - especially early in his mandate, but in the last year he's getting very political and rather more astute(his manoeuvering gets compared to Truman's, more and more).
The upcoming election should have been dominated by economic issues, which isn't the safest ground for Obama. If the right wants to interfere with this major priority of Americans and introduce moral issues, attacks on women's rights, minority rights and so on, it's not Obama who's going to complain or hinder them one bit... Heck, he will even help them when he can, strategically. More and more the Republican race gets plagued by issues of religion, moral values and so on (when most Americans seem a lot more eager to hear how they could do better than the Democrats on the economical front). Not so long ago, Romney was looking unbeatable and to have pushed Gingrich to focus mostly, but the more extreme candidates are now lasting longer than expected and back to fighting one another, and have drawn Romney into it too (and remotivated the moral right to campaign against him, throw mud at him, attack his character on the most absurb or populist things, attack him on all aspects where he's similar to Obama.. which for Obama's voters is like campaigning for Obama, when once he's the candidate, as still seems very likely, Romney will have a fairly antagonist right wing he'll need to appease and reassure, further shifting to the right, which plays right in Obama's hands. Meanwhile, the American public is watching, and Obama's advance keeps going up slowly, despite all the disappointements of his first mandate.
American women will have to make a choice later this year. 65% of them are at least to an extent pro-choice, for example. A great percentage of Americans don't let their church have much influence over their moral choices anymore or want laws to give them more power to impose choices on them, far less than the very loud and visible right wings/fundamentalists make it appear. It's largely thanks to the political structure in place that it's so hard and so long for the political leaders to push those issues to the side, as their counterparts have managed to do in most other nations, despite the fact about the same percentages of Americans would want to finally move on. Heck...98% percent of American catholic women admit they have used or keep using contraception in their life, and the majority of them have are in favour of it, and in free access to it. The Bishops have tried to get through political means what they've long stopped having any success through church/education to their own faithfuls. It's issues over which they've lost in most other western nations, often decades ago already. Catholics have no choice to contribute like everybody else to the healthcare system the majority wants, and that includes free contraception for all women who want to use it - and they've swallowed that pill and moved on. The Canadian Bishops have the same positions as the American ones on issues like contraception, abortion and so on. They're Rome's official positions. But most but the less politically astute and furthest on the right know they're better keep that for their faithful in churches (religious division in the public education system has been abolished in most of the country), or they'll only erode further the scraps of political influence that remains to them, even among their own faithfuls, that every time they managed to make the morally conservative policians make new attempts to attack those rights, it pushes the center right voters further toward the center and even center left, that from the left or (fiscal) right women unite to fight back or denounce whenever a Cardinal makes a strong public statement about these issues. The last one who did got call back to Rome (that is, called to a high standing position in the Curia) and the Bishops distanced themselves from his more extreme positions (like calling women who got raped and aborted murderers...)... So now we've got an evangelist PM who was forced to campaign on firm promises not to return to the settled "moral" issues like abortion, gay rights, capital punishment etc. if he had any hope of getting a majority.
And of course, there's a good percentage of non-Catholics among the employees of Catholic organizations too, who are the victims of this. Obama has kept his hands clean, his own position is still very clear on the issue of contraception: every American woman would want to use it should have access to it. He let retrograde, reactionary religious leaders, at odds with the vast majority of their own female faithfuls be the ones attacking, again, women's rights, again trying to impose their belief system on a majority that don't want them, in the middle of the Republican race and before the real campaign begins, and he set a nice trap for the Republican candidates to take side with them, and rekindled the motivation of the moral right to get the "right" Republican candidate chosen to be his opponent, which likely won't work but in the meantime forces Romney to address these issues when he'd rather stay in the center and try to sway Obama's voters who've found him too much to the left economically. Voters know what Obama has so far failed to deliver, and what he might prove unable to deliver even once the risk of not getting another mandate is out of the way, but they also know they won't get any of it from the Republicans because of the moral right, and what they consider their rights, and their gains, are under threat again with them - and that's not the issues they want to see debated in the next election, that's stuff they thought they had won in the last one, but obviously these issues are back again.
So yes, Obama's last compromise is disappointing and will have disappointed tons of American women and men, but now it's up to those who find this totally unacceptable to fight those battles and give him a clear mandate to return to his own positions, their own, on these matters, and on others he's failed to deliver on.
It was rather astute of Obama to "lose" strategically and throw the hot potato to the Republicans at this point, when the candidates couldn't resist take the side of his opponents, and they can't afford to just nod at the right places for the moral right at this point.