It seems clear to me that one cannot claim that acceptance of scripture is a matter of faith or requirement for Christianity. For example, the Christians of the Apostolic church had no New Testament as such, and they were surely Christians. The Greek Christians John wrote to did not necessarily even have much of the LXX (notice how in John Jesus calls the OT "your law" when speaking to the Pharisees, as though John is consciously trying to raise Christ above the parochial boundaries of the Hebrews.)
More importantly, to claim that one absolutely must accept the Bible to be a Christian necessarily means that somehow belief in Christ is no long "good enough." When the matter is put that way, it seems pretty preposterous.
A common defense is "But the Bible speaks of Christ, so refusing to accept the Bible is tantamount to refusing to believe in Christ." I wish I could say that I was just making that up. That is honestly the kind of logic so many people appeal to. By that view one could take any document that refers to Christ and make the same case. "You don't believe in IV Maccabbees? You must not believe in Christ! You don't accept the authenticity of The Book of Enoch, you must not accept Christ."
Greek mythologies speak of Tartarus, Jude also spoke of Tartarus. Does this mean that we now have to accept all the Greek mythologies because they speak of something referenced in our New Testament?
However, while belief in the Bible cannot and should not be demanded as a "matter of faith," the Christian who wants to discount it, especially the Old Testament, is in a rather difficult position. The idea that the Old Testament, at least, must be accepted as a logical consequence of following Christ seems a rarely broached topic.
Jesus regularly affirmed the Old Testament (as codified in the LXX, which was generally what was studied by Jews during that time) and quoted from 22 of its books directly. Jesus was considered (at first) as a gifted rabbi, and then later a prophet. It would have been impossible for any Jew in that tme period to have a following if there was any hint in his teachings that went against Torah. It simply didn't happen. It was perfectly acceptable to debate the interpretation of Torah, but to suggest it was wrong was unimaginable.
When Jesus is brought before the Sanhedrin, His opponents had problems finding anyone who could testify against Him. In the end, various people lied, claiming Jesus said He was going to destroy the temple. Had Jesus ever claimed the Torah was faulty or wrong, that by itself would have easily been enough to condemn Him before the Jewish council.
People point to various stories as proof that Jesus was subverting the Old Testament, but none of these are valid. Most of the time, Jesus is attacking the traditions that had built up around the Torah. An example is the washing of hands before meals. Nowhere is this required in the Torah, but it had become a practice among the Pharisees.
A particularly clear example of this is when Jesus and his disciples were walking through a grain field, hulling the grains in their hands on the Sabbath. Now, obviously there is no specific law against rubbing grains in your hand to remove the hull so you could eat it on the Sabbath. However, the Pharisees claimed this counted as "Work" and hence was not proper to do on the Sabbath. And how does Jesus respond? He defends His disciples actions by appealing to Scripture. He spoke of David's eating of the holy bread when he was in need.
Jesus reasons from the OT, quotes the OT, and constantly refers to how the Jews should have gathered from the OT all that would happen to the Messiah. When Jesus excoriates the Pharisees it is not becaue they were "legalistic." He does not attack them for relying on God's Law for salvatiohn. He attacks them for subverting and perverting God's law for their own political and personal gains. Matthew 23-23 is a wonderful illustration of both sides of this coin.