He probably is stubborn, but I suspect his problem is much deeper than that. I suspect the real issue is a spiritual problem that can be summarized in one word: pride.
Why do I say this? One reason is because pride keeps us from admitting we're wrong, or that our judgment is flawed and we make bad decisions. In addition, pride can deceive us into thinking we're better than other people, and therefore that they -- not us -- must be in the wrong. But pride also can blind us to our own faults, making us think that we're better than we really are. The Bible warns, "Do not think of yourself more highly than you ought, but rather think of yourself with sober judgment" (Romans 12:3).
But unrestrained pride is dangerous for a far more serious reason: It cuts us off from God. Prideful people think they don't need God; they're convinced they can run their life without Him. But it isn't true. It isn't true for their lives right now -- and most of all, it isn't true for eternity. The Bible says, "Pride goes before destruction, a haughty spirit before a fall" (Proverbs 16:18).
Pray for your uncle, not only that he will learn to admit his faults, but that he will realize his need for Christ's forgiveness and help. And if you have never faced our own need for Christ, humbly invite Him into your heart and life today.