But lets call it a formality, that passport, in this case a diplomatic passport would still be stamped would it not?
I do not have personal experience with a head if state but I do have personal experience with a high government official and after the hoo haa of the arrival, the “formality” is completed. Passport stamped end of story.
You are correct, I will concede, that the necessity for a passport differs by country. The Pope, being absolute monarch of Vatican City, could require a passport for himself or could dispense with one, as the Queen of the United Kingdom does. The passport is permission from a government to travel, so it is the issuing government that makes the rules for those (which is why the US government could decide that military identification is a legal substitute for a civilian passport). The hosting government decides what paperwork it requires for entry from travellers to attest to identity and permission to travel from their home countries. Obviously, a country can decide to accept travelers with no citizenship anywhere (a stateless person being someone not considered as a national by any state under the operation of its law), if they want to, just as they can deny entry to anyone they want to on no grounds at all. Each country decides what their standards are.
If the Pope were to decide not to issue himself a passport and to travel without one, he would undoubtedly be given the same right to do that as the Queen of the United Kingdom. The only question remaining, however, is whether a nation would decide to welcome him at all.
Although the Church holds that immigration is a human right, it also recognizes that every nation has the right to sovereignty over its own borders. Having standards for travel that are within the boundaries of human rights is just one of the many moral duties of a civil government, such as the duty to wage war only under just circumstances. It is possible that a government could decide that the Pope could only have permission to enter their country if he produced a passport from his country of nationality. Likewise, however, the country could also decide that he (or anyone else) needed no passport at all. Based on the precedent of the Queen, it is obvious that nations with diplomatic relations do make that concession and would undoubtedly do so for the Pope.
Having said that–I will concede that you are correct that if the Pope insisted on having his Argentine passport stamped when he traveled, the hosting nation would undoubtedly comply with the conditions he set using his authority over the Holy See and Vatican City.
I stand by my contention, however, that it is impossible for the Pope to travel as if he were just any private individual. That will be true for as long as he occupies the Chair of Peter, and for practical purposes probably always will be. He is bound by his diplomatic position; it is a fact of his life now. He is not just the Bishop of Rome but also the monarch of Vatican City and the head of the Holy See, and the latter two are sovereign entities under international law.