With some combination of "militants" and "terrorists" storming through parts of Iraq and American-trained Iraqi soldiers caving faster than you can say "Maginot Line," naturally, talk in Washington has turned to who "lost" Iraq. Republicans, never slow to pin everything from a rainy day to the impending socialist takeover on President Obama, have been shrill in their criticism. The President has been "taking a nap" according to John Boehner and was, according to John McCain, in such a hurry to leave Iraq that he forfeited the victory "won" by George W. Bush and the vaunted "surge."
The DC media, who have not met a war it did not like since Vietnam, an inchoate threat it would not mindlessly regurgitate directly from GOP talking points, or an opportunity to second-guess Barack Obama, has joined the pile on, echoing Republican assertions that something, anything, needs to be done, and please, do it fast, and make sure you do not think through the repercussions because every complicated policy decision has to be hashed out in the span of a news cycle.
While others can better rehash the litany of fuck ups, erroneous predictions and straight up lies that were told in selling the Iraq War, the fact of the matter is that the things that are roiling that country now have been known for some time - sectarian differences between Sunni, Shiite, and Kurd, endless struggles among those groups for the valuable oil reserves in various parts of the country, an authoritarian "leader" whose reflexive unwillingness to mend religious rifts that have built up literally over centuries suggested that all we had accomplished in toppling Saddam Hussein was installing a new strongman, just one who was sympathetic, not antagonistic, toward Iran. Not a small thing, by the way.
There is nothing wrong with Monday morning quarterbacking decisions that have been made, but if you are going to do that, a little honesty is probably in order. For example, the "surge" that Senator McCain is so fond of trumpeting was a lot more nuanced than simply sending more troops into harm's way. We also bribed, sorry, paid, the very Sunni militants we had been fighting to turn their guns against "terrorists" instead of us, tacitly wrote off the southern part of the country to the acolytes of Muqtada Al-Sadr (remember him?), gave a wink and a nod to nominal Kurdish independence in the north, and had little to do in Baghdad other than cement (literally, with walls) the ethnic cleansing that had largely separated Sunni from Shia. Of course, those facts don't fit comfortably on a bumper sticker or in a narrative that demands that we believe some go for broke strategy magically worked. 
As for the idea that some legacy force should have been maintained and that such a force would have made a difference when the people wearing the uniform of the home country refused to fight is laughable. The last status of forces agreement (SOFA) we signed had the name "George W. Bush" on the dotted line, not Barack Obama. When the current President sought to extend the SOFA, Iraqis refused to grant American soldiers immunity from prosecution in its courts, a non-starter that helped ensure our departure.  That fact notwithstanding, American troops left more than two years ago, at which point the "standing up" (as Shrub put it) of Iraqi forces was long past its due date. If the leaders in Iraq have done so little to engender the support, trust and loyalty of its own army, one, I might add, we spent north of $25 billion (not a typo) training, why is it our responsibility to now pull their chestnuts out of the fire?
The fact is, our record of nation building in the Middle East is woeful and the mission creep now being advanced by people like McCain and his running buddy Lindsey Graham will do little but require that we take ownership of a problem for which we will not be able to impose a solution. The sectarian battles that are playing out in places like Iraq and Syria do not end in some Middle Eastern George Washington riding in on a white horse and creating what we think of as "democracy." In Egypt, a massive uprising of students who DID want something closer to the freedom we know and love were left in the dust because the only two factions that had the organization to take advantage of the political vacuum were the Muslim Brotherhood, who won the free election for President, and the military, which overthrew that leader when the people started rebelling against him. In Libya and Tunisia, halting progress is often rewarded with steps backward that no amount of American intervention will resolve.
If what Mike Allen has reported is true, that is, that the Obama foreign policy distilled down to a bumper sticker would read "Don't Do Stupid Shit," not involving ourselves in Iraq is a no-brainer. We would be inserting ourselves on the side of a leader who thumbed his nose at us, at other major parties, and surrounded himself with a coterie of flunkies and sycophants who would not have looked unfamiliar in Saddam Hussein-era Iraq. We also cannot be expected to invest blood OR treasure into a country whose own people refuse to fight for its leader. What does it say that a band of a few thousand (at best) was able to overrun a major city like Mosul when tens of thousands of troops were at the country's disposal?
As others have noted, the first rule of digging yourself out of a hole is that you stop digging. At some point, we need to say enough - having left behind the cookbook for how to run a "democracy," we need to allow the Iraqi people to decide whether they want it. If Republicans think so little of our $600 billion a year Department of Defense (not to mention the untold billions that go to the NSA and CIA), that they are incapable of defending us in the event these extremists DO take over Iraq, what, pray tell, are we paying for now?
1. That McCain would fetishize a "go for broke" strategy is unsurprising coming from someone who picked an obscure (and unvetted) Alaska Governor as his running mate and thought it a great idea to suspend his 2008 Presidential campaign when Wall Street melted down.
2. Even if we had signed a SOFA, the residual force was expected to be fewer than 10,000 troops.