Let us all agree with Mr. Cantor and Mr. Madison here: It is indeed the case that one half of one half of one of the three branches of government should not be able to dismantle the very functions of that government over the objections of all the others. We could chalk the Federalist quote up as a case of the devil citing scripture, but that would imply a level of ironic self-awareness that Eric Cantor does not apparently have. And the president is in fact doing nothing to prevent the House from exercising their powers; the House is choosing to abandon its powers unilaterally, and in a huff.

"Seven years later, and after the nation's debt had doubled, President Obama refuses to even sit at the same table as Republicans and work to solve the 'debt problem' he correctly identified as a senator. That is a much larger failure of leadership."
Note that the nation's bills have both doubled and been cut in half during Obama's tenure, thus demonstrating just how tedious the bonds of reality are considered to be in the nation's capitol. Note that under the banner of leadership, Barack Obama has in fact sat down with House Republicans more than a little during his tenure -- the last time resulted in an agreement to implement a sh*t-laden doomsday device known as a sequester, intentionally designed as to be so farcical in implementation that even the United States House of Representatives would muster themselves as to not push the button on the thing, the most famous example of a nation underestimating their own legislators' abject stupidity since Steve Stockman was given a second chance. Note that these now-cited "debt problems" have sh*t-all to do with Obamacare, the legislatively defined House rationale for taking a lit match to all other government responsibilities. Cantor indeed continues the theme of not being able to quite pin down why Republicans have shut down the government at all, instead filling his op-ed with various pufferies that do not at all sound like a reason to crush all of government:
"Courts have held that President Obama violated the Constitution with certain 'recess' appointments, ignoring the required consent of Congress."
The consent that was explicitly withheld, typically for reasons that explicitly had nothing to do with the candidates being considered? That would seem a poor example of presidential stubbornness.
"He has abused executive-branch 'rule making' rather than working with Congress to pass laws. He has ignored the letter of the law when it comes to religious liberty and work requirements for welfare."
I swear to God, Mr. Cantor, if you were only made aware of presidential "rule making" abilities the moment Barack Obama set foot in the Oval Office, you may be a considerably duller individual than you made yourself out to be. The last president made a show of attaching something called "signing statements" to any law passed by the House and Senate that the executive branch did not personally like, statements that specifically outlined the parts of the law the executive branch objected to and stating that the executive branch would simply not comply with those portions. Perhaps all of that happened before you had made your escape from the turnip truck, Mr. Cantor, but still -- there are books on these things. Or videotapes, if that is more your style.
"President Obama has used executive orders to unilaterally change U.S. immigration laws. His administration has used waivers to change laws such as No Child Left Behind to compel states to adopt new policies."
If this is meant to be a rationale for why the majority party of the House has deemed it necessary to shutter the whole federal government, it does not come across as a very compelling one. We -- and I am using we here as general denotation of all historical adults who were not, say, Confederates -- generally do not shutter the federal government every time one branch of government does something that gives another branch of government a bad case of the sads. As someone on the liberal side of things, I could give you stories of things that happened during the last few administrations that would seem far more compelling indications of an executive abuse of power than "he hath given waivers to an education law that we doth not liketh very much" would seem to be. I believe you personally stood in support of a good number of them.

My only interpretation is that Cantor is no longer sure himself what he is trying at:

"Just two years ago, President Obama and House Republicans came to an agreement to raise the debt ceiling and implement much-needed reductions in spending."
...which would seem to further dissuade from the Obama-as-tyrant theory, a true shame after holding up such convincing examples of tyranny as executive "rule making" and allowing "waivers" on things. (Republicans are generally keen fans of states receiving waivers to exempt themselves from federal laws, but anyone looking for Republican ideological consistency during this particular decade has likely shot themselves by now.)

No, it seems the impetus here was that the House majority leader was expected to write an op-ed because Paul Ryan had done one too, or perhaps merely has hastily constructed shield against public opinion polls that now have approval of House Republicans hovering somewhere between chewing broken glass and infecting yourself with the Black Plague. His message is that House Republicans are being disrespected, the Senate is non-entity not even worth considering in a purely rhetorical context, and Barack Obama is a tyrant so brutal as to require Republican rebellion against his rule. 

None of it suggests any specific demand on the part of Republicans other than the general demand that Obama give a generic something as token of House superiority. None of it suggests that the majority leader of the House gives the slightest damn about the sweeping hardships they have inflicted on the nation, so obsessed is he with demanding that abstract something from the president. The party of white southern men has got a monumental bug up their ass about how they are not being respected by the first non-white president ever given the ability to make executive nominations and waivers and rules and by their God, they are not going to reopen this government until he learns his place.