How does someone come to accept Biblical Authority?

The underlying question here is ultimately, "How does someone become a Christian?"  This is tied in with irresistible grace .  Unless the Holy Spirit radically transforms the heart (John 3:6-8), a man will never believe the gospel and therefore accept it as the absolute authority in his life.  For this reason, our evangelistic approach must always be one that asserts the authority of the Bible, for this is the means by which the elect will come to know God (Rom 10-14-15). 

We reason through the text, but we do not try to prove its validity based on terms set by the non-believer.  One who presupposes a world without God is only going to be convinced that He exists through a radical work of the Spirit through the preached Word and not through evidences about God.  These evidences have their place, but they must be used in the way they were intended. There is a difference; take special notice of what Paul is doing here in Acts 17 and look at the reaction of the people listening.  We will look at his handling of the Jews and God-fearing Gentiles first.

¹Now when they had traveled through Amphipolis and Apollonia, they came to Thessalonica, where there was a synagogue of the Jews. ²And according to Paul's custom, he went to them, and for three Sabbaths reasoned with them from the Scriptures, ³explaining and giving evidence that the Christ had to suffer and rise again from the dead, and saying, "This Jesus whom I am proclaiming to you is the Christ." ⁴And some of them were persuaded and joined Paul and Silas, along with a large number of the God-fearing Greeks and a number of the leading women.  ⁵But the Jews, becoming jealous and taking along some wicked men from the market place, formed a mob and set the city in an uproar; and attacking the house of Jason, they were seeking to bring them out to the people. ⁶When they did not find them, they began dragging Jason and some brethren before the city authorities, shouting, "These men who have upset the world have come here also; ⁷and Jason has welcomed them, and they all act contrary to the decrees of Caesar, saying that there is another king, Jesus." ⁸They stirred up the crowd and the city authorities who heard these things. ⁹And when they had received a pledge from Jason and the others, they released them. Acts 17:1-9

Paul reasoned with the Jews and God-fearing Gentiles through the Scriptures, and He did so with them at a deeper level because of the knowledge they already had of the Scriptures.  Nonetheless, he did not appeal to human philosophic reasoning but to Revealed Scripture.  There were those who believed and then those who did not, but their belief did not come from Paul trying to use human philosophic reasoning.

Likewise, Paul does the same thing with the Gentiles on Mars Hill.

²²So Paul stood in the midst of the Areopagus and said, "Men of Athens, I observe that you are very religious in all respects. ²³"For while I was passing through and examining the objects of your worship, I also found an altar with this inscription, 'TO AN UNKNOWN GOD ' Therefore what you worship in ignorance, this I proclaim to you. ²⁴"The God who made the world and all things in it, since He is Lord of heaven and earth, does not dwell in temples made with hands; ²⁵nor is He served by human hands, as though He needed anything, since He Himself gives to all people life and breath and all things; ²⁶ and He made from one man every nation of mankind to live on all the face of the earth, having determined their appointed times and the boundaries of their habitation, ²⁷that they would seek God, if perhaps they might grope for Him and find Him, though He is not far from each one of us; ²⁸for in Him we live and move and exist, as even some of your own poets have said, 'For we also are His children.' ²⁹"Being then the children of God, we ought not to think that the Divine Nature is like gold or silver or stone, an image formed by the art and thought of man. ³⁰"Therefore having overlooked the times of ignorance, God is now declaring to men that all people everywhere should repent.  ³¹because He has fixed a day in which He will judge the world in righteousness through a Man whom He has appointed, having furnished proof to all men by raising Him from the dead." ³²Now when they heard of the resurrection of the dead, some began to sneer, but others said, "We shall hear you again concerning this." ³³So Paul went out of their midst. ³⁴But some men joined him and believed, among whom also were Dionysius the Areopagite and a woman named Damaris and others with them. Acts 17:22-34

Though his audience is probably all Pagan Gentiles here, Paul still uses the Scriptures to make his case, for he is constantly alluding to who God is and our need to repent, and He does so on the basis of Revealed Scripture.  The difference in approach is the depth at which he goes.  He does not water down the gospel; rather, he explains it a clear level by meeting them where they are at in regard to their knowledge of the One True God, which is discovered through the Scriptures.  

Therefore, presupposing the authority of the Scripture is the right way to evangelize and disciple.  Extra biblical evidences show how the Bible corresponds to reality perfectly, yet is the Holy Spirit working through the Preached Scripture that will ultimately cause someone to be born again, thus enabling them to repent and believe in Christ for salvation.  It is then that they will see how it is the Scripture alone that makes sense of all these evidences.  This is why our theology of conversion is important; because we believe that conversion is an act of grace and cannot be conjured up by the will of man, then we must appeal to God's ordained means in order to achieve His ordained ends.  His means is through preaching the Scriptures while presupposing Biblical Authority, and this achieves His ends, which is the electing those whom He chooses unto salvation. This is clear in Acts 13:48,  "⁴⁸When the Gentiles heard this, they began rejoicing and glorifying the word of the Lord; and as many as had been appointed to eternal life believed."

To Him be the glory! Amen!