While I am not as familiar with the Muslim perspective, from the Christian perspective the issue with homosexuality never came up with Jesus the Christ nor was any quote on it attributable to him.
The Old Testament or "First Covenant" of what Christians put forward as the bible was designed to gather a people of God, a nation. In the case of the First Covenant, following the principles and regulations contained in those books would bring one into relationship with God and is also known as Judaism. Jesus was a Jew; he was not a Christian. He followed Jewish law as contained in the First Covenant but told his disciples that he came to "fulfill" the precepts and regulations with a "New Covenant"; hence the "New Testament." The New Covenant was in response to layer upon layer of regulations being heaped upon normal persons by religious of the day (I believe it was more than 500 "things" that had to be done in order to be in covenant). Jesus' message was simple: love God with all your heart and soul and love your neighbor.
For me, after studying and readying for ordination, I am saddened by the way "Christians" try to pick and choose from both sections of the bible. If you want to pick from Leviticus and the Old Testament as desires and things that will be followed then you are not a Christian but should more properly join the Jewish faith. The Old Testament is not a literal history or biography; it was designed to lead a people into a relationship with a God which had very specific references and tenants that, if followed, assured that the person would achieve a covenantal relationship. It was largely based on rituals and sacrifices -- repeated over and over.
The New Covenant offered a covenantal relationship through the Son of God or Jesus the Christ. That covenant was simple, as I stated earlier, love. It's hard to do anything negative or ill to someone if you truly have an all encompassing and true love. Jesus did not hang with the religious of his day; he usually reserved his harshest words for them. Who did he hang out with? Prostitutes, tax collectors, divorced, sick, criminals, and fishermen. If one truly believes what is written, he was tempted in all things as we are but was found to be without sin. As my confessor/priest told me when I decided to forgo ordination: would he have been tempted by homosexuality as well? And if God creates all things in his image and homosexuality is not a learned trait but innate, where does that put an all-perfect God?
Thus a Christian is to follow the New Covenant (or Testament) writings in which Jesus says nothing about homosexuality (but does say a lot about divorce which is largely ignored). His disciples, primarily Paul, have the word "homosexual" attributed to them but there is no direct word in Greek that matches and homosexual behavior was quite the norm. Somehow I believe Jesus the Christ would have been at the Pride parade; he would have been helping shelter the gay thrown out by his parents and beaten on the street; he would have healed the person with AIDS/HIV.