I would say to me it is obvious that homosexuality isn't a disease and therefore can't be cured, but it is scary there is someone making this claim right now, and being taken seriously as well. I am glad you decided to address this, and challenge the assumption instead of ignoring it.

The level of division this would create would be incredible. i think yeah, it would be compulsory in some countries, particularly ones with a national religion. The weapon it would be for the hate mongering of some evangelical churches would be phenomenal, given how they actively encourage the discrimination, incarceration and psychical punishment of gay people in other countries already... alright then... a less cynical prediction this time haha

So i think, if they found an injection that suppressed same sex attraction, a lot of things would happen.

• firstly when this was reported in the news as a 'cure' (as the person you encountered described it) , there would be outcry just around the language, but i don't think that outcry would stop it making it to market. When it came to making this widely available,i believe there would be protests on both sides, the hate would intensify between the ends of spectrum. The gay community would be split into three main (and obviously loads more but being broad here) factions.

1. The 'cured' - those who are actually happy about this, these would be the minority (but probably also would be projected into the spotlight by people who wanted to make the injection compulsory - and also likely a decent amount of people in this group would secretly belong to the next).
2. the 'compliant' - those who do not want to take it, and feel like they have to, due to peer pressure, social pressures, their own community/family etc etc and end up giving in our of fear,
3. the 'resilient' - those who won't take it, and resist being forced.

• I believe this would be rolled out on a voluntary basis to begin with. There would be a huge amount of division, as people felt like they had to chose sides, but were scared to say anything - rather like you have in your comments haha, and as a result it would slide past a tipping point of hate on both sides before most people got involved.
  I suspect there would be riots, and protests outside clinics/facilities that offer the injection, from both sides. The 'don't take the shot' side of the protest would try and convince people not to take it, whilst the 'take the shot' side would try and get between the two. Rather like some of the show downs outside abortion clinics in the US, it would get very heated. I suspect, given the nature of the issue, and how marginalized parts of the gay community already feel, this would lead to extremism, and we would see physical acts of terrorism on both sides.
  As that aspect escalated, it would allow one side to demonize the other, and the general climate of the country would determine which way that went. More religious countries would be more likely to paint the 'don't take the shot if you don't want to' side as the bad guys, and the 'take the shot' as the hero's trying to stop them, but i have to believe it would be the other way around in some places.

• once we were at the extremism point, we would probably already being seeing forced injections, probably in religious areas. Those who wanted the injection given to everyone, would claim an association between peoples sexual orientation and the lengths they have gone to defend their freedom (violence, protests, etc), and use that to further smear all gay people. They would take the worst behavior of a single defender of not taking the injection, and apply it to all gay people, they would use it as leverage to allow for discrimination.
  I suspect certain private schools would require a child to be 'vaccinated' to go there, and the fear mongering around the protests and objections would lead to more parents considering injecting their own children - for their own safety of course. I feel like phrases like 'gays are violent don't you know', and 'they are sick, they just need help' would become prevalent.
  Some charities would only grant overseas funding if the vaccine was made compulsory and private organisations would be able to easily use it to discriminate on the grounds of safety. In certain areas, it would surely lead to an increase in homophobia, which would in turn lead to an increase in people both wanting to take it for the wrong reason, and people being forced to take it. I think a surprising amount of people could end up taking it.

• I suspect as it became more harsh in some places, it would soften as a reaction in others. As people in one country saw violence increase, people resisting the 'vaccine' until the last minute, and the real life horror stories surfaced, it would lead to other countries/places going the opposite way. In countries where it was maybe adopted but not enforced, I suspect the amount of people who flee as refugees due to persecution would massively increase, combined with those fleeing to avoid being forced to take the injection, and those moving to live in a 'cured' zone, it would lead to a decent amount of global relocation. As things progressed, what remained of the LBQT+ community would fracture, those wanted to continue to live as normal people, and those who's hate had twisted so badly. There is already A LOT of really vicious cishet hate in the gay communities. These people who already passionately hate straight people may splinter off, and form a society of their own but i suspect the sheer hate they already carry would be more likely to lead to terrorism. Just as I suspect the hate on the 'take the shot' side would likely lead to extreme acts of violence against individuals.

• in the short term here, i feel like the hate on both sides would get to levels that supersede anything we have seen in history. However, i feel like the long term result would be mixed. Not all children would be vaccinated, and out of those those who weren't, some of them would experience their natural attraction. I suspect, given that their parents chose not to vaccinate, those children who did not get the shot and came out as gay, would be celebrated by their families - and the communities their families had chosen. Once we reached the point of 'next generation' homosexuals , the scales would tip. The children who had had it forced on them would feel like they had something taken away from them, there could possibly be a rise in asexuality in those children as they felt it had been taken away from them. I think rather like any historical persecution has given way to a much wider acceptance in the very long run, the next generation would celebrate those who stayed true to themselves, and those who had been forced to take it as babies would be the ones who outlawed forcing it on other babies.

I don't know if it is too hopefully to think that such horror would change us. I can't see how that wouldn't lead to horror, but that is not to say that horror wouldn't lead to a massive social change at a later point. The historical playbook i would base speculation on, would be the conflict in northern Ireland, which is now leading to a generation of young people who want nothing more than to accept each other as they are, for who they are, regardless of their beliefs or practices. Hate begets hate, which begets hate, which in the end, begets a hatred of hate. Its the repeat or repel cycle of abuse, eventually the chain snaps, and it snaps hard the other way. I hope.

Update: Then again, I like to think that maybe, just maybe, people would be better than they have been in the past, and it wouldn't be forced on anyone, nor would it be given out easily. I like to think, if they did find something that would make people not homosexual, that taking it wouldn't be something people did lightly. That if someone went to a doctor and asked for it, they would get counselling, and have to undergo a wait period, rather like how in the UK the first step in a sex change is normally two years of counselling preparing for the change, and making sure it is what a person wants. Maybe, just maybe, it would go like that, and the 'vaccine' would be given sparingly, and only to those it would really help. Jeez, given how there are some people who are 'reformed' homosexuals in the church, and repeatedly 'relapse', i can see there being a group of people who desperately, desperately don't want to be attracted to the same sex, but can't fight it. Maybe, maybe just maybe, it would be something good for them, and not forced on anyone, or given to anyone who may not be 100% about wanting it.