Did Jesus approve a same sex relationship?
That's how Today's Gospel has been interpreted in contemporary gay exegesis, based upon translations and meanings of the original Greek words, pais and doulos - used for servant or slave in different accounts - which can have a sexual connotation. A Roman might have a slave used for sex, or a young male concubine. The interpretation makes sense, but I'm not sure the Fathers would have understood it exactly in that way. Even if they had, homosexual acts can never be approved, therefore it could never be interpreted as approval of a gay union.
Regardless of how it is understood, it's absurd to imagine Jesus giving his blessing upon a same sex relationship in either account from Luke or Matthew. The Centurion prayed for a cure for a beloved servant-slave, no mention is made of their relationship or domestic situation. Some suggest that the Centurion didn't want Jesus to enter under his roof because he was afraid the young servant would fall in love with Jesus and leave him. Where do people get this?
When Christ said he would come and heal the servant,
The centurion said in reply,"Lord, I am not worthy to have you enter under my roof;
only say the word and my servant will be healed."
We say that before the reception of Holy Communion.
It is an acknowledgement that we are unworthy to Have Jesus enter under our roof, our soul. It is an act of faith, a sort of confession of sin, and/or of our unworthiness. It can't be a pretense or something to hide behind to feign humility. The faith of the Centurion in the power of Christ is detailed by the soldier as he describes how he commands those subject to him and attributes the same power of command to Christ. Our Lord healed the servant because of the Centurion's faith. What their relationship was is never part of the narrative. Perhaps it was a homosexual relationship - how would anyone believe from the narrative that Jesus approved of it? In the case of the Samaritan woman, he didn't show approval for her relationship with a man who wasn't her husband. neither did he approve of the sins of the woman caught in adultery - he told her to go and sin no more.
If anything, Today's Gospel demonstrates that salvation is available to the Gentiles, even those in sinful relationships, and Jesus shows us once again that His mercy is for everyone. Perhaps the Centurion didn't want Jesus to come to his house simply because he was ashamed? Maybe he really meant what he said, "Lord, I am not worthy that you should come under my roof." Maybe when he returned home and found the servant healed, they both became followers of Christ? That's a more reasonable interpretation. So imagine what you will, Christ came to call sinners, so it doesn't matter if the Centurion and his servant were straight or gay. When we follow Christ, we leave behind our former way of life, we repent of our sins.