Actually, Protestants have the harder time making their case, because as one convert to Catholicism points out, nowhere in Scripture is it written, "Sola scriptura." Also, as John Donne remarks in one of his sermons, the first Christians had to ask the Jews, "Which books were the scriptures?"
Catholics believe that God guides His Church to the Truth, and therefore when the Church decided that certain books were canonical and which were not (effectively decided at least as early as St. Jerome but not formally confirmed until the Council of Trent), they had divine guidance. Similarly the writers of the various biblical books were divinely guided.
There is, of course, no way to prove this unless you have faith, something Protestants and Catholics agree on. But I would find it hard to study history and not conclude that God intervened in history, first to the Jews and then to Christians. Both the Jewish and Christian stories are firmly anchored in history, although it's possible to be a skeptic and disbelieve it.
The thing is, "sola scriptura" and Biblical literalism are two separate things. I am what you would call a Biblical literalist but I don't accept sola scriptura. Unfortunately, many Catholic/Orthodox anti-literalists like to obfuscate the difference between the two.