Well, we've had a new Pope, in Pope Francis I, for over a full day now. Not we, as the Catholic Church, but we as the world. I say the world because no matter what your religion might be (or if you have no religion like me) the Church is still very powerful in much of the world. Heck, we even refer to it just as, the Church.
Now, the Catholic Church is not like the Hare Krishnas, a group with at best a hundred thousand people in the U.S. The Catholic Church is a very dominant force in politics and culture. In Japan politicans don't have to worry about what the Pope might be thinking, but when he speaks in the U.S., (or even an influential Cardinal like Timothy Dolan of New York) the media reports. It is news.
In even more religious and Catholic nations like Mexico, the Catholic Church is even more influential in national and local politics. Of course, this influence isn't absolute, that is why in deeply Catholic Argentina (the place where Pope Francis I is from) there is now marriage equality for gay couples. But in California, the Catholic Church along with the Church of the Latter Day Saints (the Mormons) funneled big bucks into defeating marriage equality. At least for now.
When it comes down to it, the Catholic Church and the Vatican is only a religion. It has almost no land, the only people who live in the Vatican are employed by it in some way. Saudi Arabia, a horrible Islamic theocracy, at least has actual land and people. So this is what I suggest to President Obama, end full official recognization of the Vatican/Catholic Church as a nation state.
The Catholic Church/Vatican as a nation state dates back to a deal with then Italian dicator Benitio Mussolini. I doubt though there are any statues of this dictator in the Vatican. Full recognization of the Vatican by our nation only dates back to the mid 1980s, so it isn't like it has a long tradition.
Taiwan, a democracy with millions of citizens, doesn't have an American ambassor or embassy, though that has to do with the influence of China. The U.S. doesn't have an embassy to the Church of England or Sunni Muslims or Southern Baptists or Orthodox Judaism. Now, I know President Obama would suffer a heap of abuse for doing so, but sometimes, you have to face backlash to do what is right. We are a secular/Enlightenment based republic, we shouldn't the Vatican as an equal, they are not.