*Disclaimer: I or PFC do not necessarily agree or disagree with the above conclusion of this meme, this exercise is merely a look at the use (or lack thereof) of critical thinking.
For starters, the nature of internet memes is such that there is almost inevitably going to be a massive case of the Straw Person fallacy going on. There are in fact many quite nuanced arguments against H, drawing inspiration from the Bible that cannot be so casually dismissed with a sentence, more work should be done.
Now working from left to right.
"Because Jesus Said So" - the objection is true, although a case could be made that inferences could be drawn from what he did say on similar topics.
The first blue box, "Have fun..." unnecessarily insults people who hold to a certain view. This is known as an ad hominem attack and in this case bullies people into agreeing with the author rather than actually engaging in debate.
The next two boxes, "OT and NT" end with false dichotomies. There are other ways to answer the secondary questions than just a simple yes or no (although flow charts get messy when they have to deal with every possibility).
A theological insert here - to say that the Bible clearly defines anything can often be a stretch, and both boxes in this route fall prey to a rather shallow reading of the text.
The appeal to the "ick factor" in the final route is an interesting one and should not be so quickly dismissed. Yes we should be tolerant, and there should under no circumstances be hateful discrimination. However, there is something to be said for prima facie arguments, appeals to the obvious norm (this is basic anatomy). Much can be said for alternatives, but the simple fact of our anatomy remains and questions can and should be raised about a cavalier approach to sexuality: Are all practices healthy, safe, beneficial? These issues should be examined in and of themselves, not simply dismissed out of hand.
Finally, the only green box, the alleged "right answer". The inference of this comment is that those who disagree are uncivilized barbarians, which is simply not the case. There are civilized, educated people with very real concerns and to paint them as something they are not, is as bad as discriminating against homosexuals. Fighting discrimination with discrimination, while perhaps rhetorically effective, is not logically sound.
In conclusion, this meme is riddled with fallacies and logical shortcuts. Complex questions very rarely have easy sloganesque answers.