Under what criteria were these books accepted?Who first accepted them? Did Christ or the Apostles ever make a spefic reference to these books?
There are no direct quotes, by Christ or the apostles, of the OT Deuterocanonical books, but that is not a criterion for accepting their canonicity, as there are other OT books accepted by Catholics and Protestants alike that neither Christ nor the apostles quoted from, e.g., Esther.
The way a Catholic accepts any book as being inspired, whether NT or OT, is only by the authority of the Catholic Church, as there is no other method to know with certainty which books should be found in the Bible. That’s one reason why Catholics do not believe in Sola Scriptura.
If Sola Scriptura were true, then the Bible would be the sole method of gaining the knowledge of which books should be in the Bible, but the Bible does not contain that information. And even if it did, one would first have to arrive at the conclusion that the book where that information is found belongs in the Bible, which is circular.
The canon of the Bible comes from oral tradition only, and as between Catholics and Protestants, the question becomes: Whose oral tradition is correct? However, believing in an oral tradition contradicts the very basis of Sola Scriptura. By what method does a Protestant come to the conclusion that there are 66 books in the Bible? The Protestant must show chapter and verse, and use the Bible only, if he/she wants to adhere to Sola Scriptura. Otherwise, he/she is merely taking someone’s opinion on the matter.
The question is: What is
God’s opinion on what constitutes Sacred Scripture? Using the Bible alone, which is the only place a Protestant can legitimately go for the answer, how does the Protestant answer that question?
In any event, no one has a complete Bible, not even the Catholic Church. Some parts of Sacred Scripture are missing. For an example, see 1 Corinthians 5:9 (KJV): “I wrote unto you in an epistle not to company with fornicators.”
That statement is from Saint Paul’s so-called first epistle, but it demonstrates that there was a prior epistle written by Saint Paul to the Corinthians.
Also see James, chapter 4:5 (KJV): Do ye think that the scripture saith in vain, The spirit that dwelleth in us lusteth to envy?
That quote is found nowhere else in the Bible, but Saint James calls it Scripture. Therefore, there is more to the Bible than what we have, which I would think would be a problematic thing for those who believe in Sola Scriptura. Their sole rule of faith is incomplete, so how do they know what else they should believe? What is contained in those missing inspired books?
May God bless you in your search for Truth.