The first thing to consider in this question is Jesus’ background. Though he lived in the larger Roman Empire where sexual morals were as lax as any in modern America, he also lived within a specific culture within the Roman world, Judaism, and that shaped him far more than Greco-Roman thought. First century Judaism had a different sexual ethic from much of the Roman world. This ethic was grounded in the belief that God had spoken through the prophets and revealed the purpose of human sexuality and that we are to control those sexual impulses which lie outside of that design.
The Gospels never record Jesus speaking on homosexual behavior. [Which is the only category 1st century people had. Meaning they did not think about sexual identity they were only concerned with the physical act of sex not with sexual attraction.] It is very dangerous for us to attempt to construct an argument from silence which would cut across the tradition Jewish belief. In fact, Jesus teachings which have been recorded are typically clarifications of disputed passages, or fundamental re-orientations of Jewish thought. Plainly said the Gospels only seems to record those moments when Jesus is in conflict with the Judaism of his day. If this is the case we must be more likely to assume that Jesus agreed with the Jewish ideas prevalent at the time which banned homosexual practice. Some might try to argue that this teaching would be so scandalous that the Gospel authors simply omitted it. But, this argument does not hold up since the Gospel authors recorded teachings which would have been at least equally scandalous (and perhaps more so). Also, the one time Jesus is presented a question about sexual morals (Mark 10:2-12, Matthew 19:3-12) he actually takes a stricter approach than most Jews of his day. Jesus argues that God instituted marriage for specific reasons and we as created beings do not have the authority to undermine those reasons. Some will point to John 8 to show Jesus’s compassion for those who violate the marriage vows (and thus attempt to take the line that Jesus was more lax on marriage); however, this line of thought does not take into account the situation Jesus was confronted with, nor his response to the woman, “Go and do not sin. Clearly Jesus was not condoning her actions rather he was offering her a chance to turn her life around and live as God designed. We must also give some credence to the writings of Paul (the only New Testament author to discuss the issue). Paul was usually very faithful in distinguishing between his own customs/practices/beliefs and those which need to be adhered to by the larger Christian community. Paul mentions homosexual practice in a few places (again Paul would not have understood what we mean by “homosexual” it was not an idea then), and he lists homosexual practice alongside slander and murder, as sinful behavior resulting from selfishness. Paul’s use of homosexual practice (see Romans 1) requires detailed study and cannot simply be glossed over, Paul’s comments need to be explored and understood in their own right. But I address them here because Paul makes no qualification about his beliefs, had Jesus given a differing teaching or an ambiguous teaching on the issue we can be sure Paul would have noted that. Paul would not simply have left an uninterpreted statement on an issue which Jesus touched on unless he was in complete harmony with Jesus. There are Christians who would argue that Paul’s words are very culturally specific and that the Bible as a whole is silent on modern homosexual practice. If this is the case, then we have more freedom to interpret our own moral construct on the issue of homosexuality. However, it is highly questionable whether these scholars meet the burden of proof necessary to adjust Christian understanding (and yes in this case as with many others the burden of proof would fall on those seeking to change the consensus).
At this point it seemsnecessary to make two addendums to this question. First Christians never discount or seek todestroy the love between two individuals. However, it is irresponsible to assume that love and sexual relationshipare the same. Simply because twoindividuals love one another does not inherently allow a sexualrelationship. To define the legitimateboundary of sex as “love” fails to understand what love is. Love can be expressed in a myriad of ways andin a myriad of relationships. Mostpeople can agree to this, and further most people would recognize there areseveral types of loving relationship in which sexual expression isinappropriate. The current argument overhomosexual expression is not about who loves whom. Within the Church the discussion is aboutwhat did God mean in the definition of marriage and what is/are the purpose(s)of marriage. Clearly the Bible does notconsider “love” on its own sufficient grounds for sexual activity (or if itdoes it is not simply the love between the people engaged in the sexual actbecause some sexual acts have extremely large consequences, i.e.adultery). This is a very difficultissue which is grounded in Biblical theology, but which gains insights fromother areas such as biological, neuroscience, psychology, anthropology, andeven zoology. All of this to say it is avery complex issue.
The second addendum is that simply because Jesus did not speak on the topic and that Paul seems to consider homosexual practice sinful does not mean that Christians need to be belligerent about the issue. Jesus and Paul both warn believers to be humble and to considerate with others even when dealing with their flaws. Meaning that we must be very considerate of others and individuals carefully not simply expecting people to toe a line. As one of my professors put it we all have monkeys on our backs and we must be aware of that fact even if your monkey is different from my monkey.
For more detailed info see first Mark Yarhouse Homosexuality and the Christian: A Guide for Parents, Pastors, and Friends
Then Douglas Moo Romans New International Application Commentary.
Quite honestly so much of the literature on the topic is so polemic it is hard for me to recommend much. There is a world of literature on all sides but so little of it is very helpful or else it is very technical.