"Because many Christians assume that those who support same-sex relationships do so out of a desire to satiate their appetites rather than sincere Christian convictions ...."
Whatever one's motives for promoting what the overwhelming witness of Scripture regards as high immorality, one is misguided. One can be misguided even from what seems like good (but really ill-informed) motives.
"I'm as single as ever and have remained celibate throughout my twenties .... But it feels important for me to … support people in same-sex relationships so that friendship is promoted as a good in itself rather than a quick fix for the gay problem."
There is nothing wrong with promoting friendship as an alternative to committing the immorality of homosexual practice. A good is not any less of a good because it is a God-honoring alternative to a God-dishonoring behavior. Friendship also is not "a good in itself" but a good only if it promotes God-honoring thoughts and behavior.
"My goal has been to help Christians create the kinds of communities that make LGBT people feel wanted …. That looks a little different to me now that I've seen so much fruit in affirming communities, but it's a widening of my circle — not a move in a different direction …. It's not too late to call it quits on all the fighting."
One's ultimate goal should not be to "make LGBT people feel wanted" but rather, as Paul says, to "wage war" not in physical battles but in spiritual ones, "tearing down rationalizations and every high thing lifted up against the knowledge of God, and … taking captive every thought to the obedience of Christ" (2 Cor 10:3-5). That is the kind of fighting that the gospel calls us to do.
One cannot change the commands of God in a vain hope to make violators feel more wanted. We are wanted because God pursues us in love in spite of our sins and in order to take us out of the same sins. Jesus and the apostles often did not feel "wanted" in doing the will of God. Our goal is to do the will of our heavenly Father and to seek first his Kingdom. Our goal in ministry is to conform people to the image of Christ (Rom 8:29), which can never be achieved by granting a license to disobey central moral commands of God.
Unfortunately, Julie has deluded herself into thinking that she is "not moving in a different direction." Promoting homosexual relations is an anti-Christian direction. "Fruit" can be manifested even in the context of adult-committed incestuous or polyamorous relationships. That doesn't make such relationships acceptable to God. The good that one does in the midst of evil does not sanctify the evil.
"If it turns out that I'm wrong, I trust God will be faithful to catch me. For now, though, I hope those of you who disagree will continue to welcome my friendship and serve alongside me. We could choose instead to focus on all we share in common and seek to mend what's been broken in this fragile world."
To promote immorality while expressing the conviction that "God will be faithful to catch me" is the kind of cheap grace against which Dietrich Bonhoeffer warned, based on numerous warnings in Scripture. Friends reprove friends when they veer off into unfaithful conduct or encourage others to do so.
Christians cannot serve alongside another who is promoting gross violation of God's commands. What Christians "share in common" is the confession of Christ as Lord, which confession requires a life of obedience rather than rebellion. It is precisely because homosexual practice breaks what God is mending that we cannot be part of Julie's new endeavor.
With Jesus we say: "Go, and from now on no longer be sinning, lest something worse happen to you" (John 5:14; 8:11).