Can a Christian struggling with same-sex attraction live a happy and fulfilled life?

When it comes to questions of ‘happiness’ we need to ensure our views are shaped by God’s word rather than our culture. Sex is not essential for a fulfilled life. Our Saviour was single and celibate and the Apostle Paul was also single. The lie of our culture, that you need sex or a romantic relationship to be ‘happy’, is untrue. Being unmarried is also celebrated in the Bible. The FAQ ‘Can a single person be as happy as a married person?’ touches on this.

Why would God make someone gay and then deny them a loving, committed same-sex relationship?

This question assumes that God made someone gay. So the first point is that it's not been proven that people are born with same-sex attractions (see FAQ "What causes same-sex attraction?"). Secondly, the question ignores the biblical truth that we live in a fallen world where every one of us has desires that God calls us to deny.