My question is, is that really what they call him? "The Holy Father"? Or was that a slip of the tongue?
It was probably intentional, but Roman Catholics do not use it to imply that this, or any other Pope, is God. The word "pope" derives from the same greek root that gives us "pappa", and referred to a bishop. The Pope is the Bishop of Rome; Eastern Orthodox bishops are "patriarchs", as in the Patriarch of Constantinople. A bishop is the "father" of his church, the "shephered" of his flock.
The Bishop of Rome is a special case, and how he got to be special is a matter of historical controversy. Before about 600, the supreme authority in the Church was a church council, which could only be called by a Roman Emperor. The Bishop of Rome was considered the senior bishop, because (a) his diocese had been founded by St Peter, and Christ had made that pun (Petrus = rock) "on this rock I shall found my church", with the power over the keys to heaven, etc. And (b) Rome was the first capital of the Empire.
The Eastern Orthodox consider that the "keys to heaven" are given to all bishops like Peter, to the Church in general. Furthermore, after around 300, the Empire had two Emperors: one living in Rome, speaking Latin, and one living in Asia Minor, speaking Greek. As Constantinople became more powerful -- cities, population, trade, and civilization were all stronger in the eastern Mediterranean -- the Bishop / Patriarch of Constantinople became more powerful.
If you followed argument (b), then the Patriarch was more influential than the Pope, although Orthodox were likely to say that the Pope deserved special honor -- not power, but honor -- as the senior bishop.
Between 640 and 1100, (a) Arabs conquered most of the other cities that had bishops who derived their power from the original set (Jerusalem and Alexandria, for instance), and (b) officials working for the Pope began to claim more and more authority for him. In part, this was an effort to establish a central power center that could reform the Church along the lines proposed by people like St Benedict: the clergy should be celibate, should shave their beards, etc.
The Eastern Orthodox, in general, said, "Hey, we never agreed to that!", but they had enough trouble fighting Arab and Turkish armies. There was a bizarre incident in which the Patriarch and the Pope mutually excommunicated each other, but you should read about that in Sir Steven Runciman's "The Great Schism".
I don't want to do a sloppy summary of a carefully researched and contentious history.
Summary:
No, Roman Catholics do not believe that the Pope is God. By slow evolution, the Church has come to believe that in certain special situations (when speaking "from the chair", I believe) the Pope's message is "infallible", but I don't see that there is an easy and clear guideline on when the Pope should use infallibility. Certainly, the next Pope will not wake up one day an declare the law of gravity null and void.
Popes today have more dignity and respect, but less political authority than they did in, say, the time of Dante.
