If you tell me you truly believe that 2 + 2 = 5, I will tell you I truly believe you are wrong. If you persist in believing this equation to be true, I'll probably continue to hang out with you and go to the movies with you and be your friend. You'll just be wrong about math and I probably won't want you to be in charge of teaching it to any children. I'll be sad about your incorrect math skills. But life would go on and at no point would we hate each other over this disagreement.
That's a simplistic view of conflict over deeply held beliefs that does not end in fear and hatred. I'm not sure why we can't apply it to other things. For instance, I believe abortion is wrong. The act itself is intrinsically evil, which means it can never be justified. However, this does not mean I hate women. It does not mean I hate women who have had an abortion. Identifying something as a sin should not actually lead to hate and fear of the person. As we're called to bring Jesus to people and Jesus came to save sinners (like us), we should continue to love these people. This brings about the "hate the sin and love the sinner" catchphrase that is often thrown around but oftentimes not understood or put into practice.
Of course, some might argue the analogy to math isn't valid as we know math to be true. Some would argue we don't know that about God and morals. I'm an engineer, I like math. I trust math. However, I believe in God more than I believe in math. "Faith is certain. It is more certain than all human knowledge because it is founded on the very word of God who cannot lie. To be sure, revealed truths can seem obscure to human reason and experience, but 'the certainty that the divine light gives is greater than that which the light of natural reason gives'" (Catechism of the Catholic Church 157, quoting St Aquinas). The analogy is valid for a believer.
Now let's go back to the math. Let's say you were trying to write laws that allowed the teaching of your 2 + 2 = 5 equation. I would oppose your laws. I don't think it will bring good order to society to teach such things. I'll probably invest money in the campaigns against your campaign and put a sign out in my yard. However, this still would not prevent me from hanging out with you, assuming we could remain civil. It is always harder to remain civil with politics, but that would be a goal.
So when there are laws in favor of abortion, I must also oppose them. Naturally if I believe in God and think He's opposed to such actions, I wouldn't want to support laws that allow them. I'd want to invest in campaigns that are opposed to these laws and hopefully participate in educating people as to why abortion is wrong while providing other alternatives. Again, this does not mean I hate women. It does not mean I hate women who have had an abortion. I don't sit around at night wondering how I can make life harder for another woman while laughing maniacally.
When the Catholic Church or others identify behaviors as sinful and when they fight policies that promote sinful actions, they are not also hating on their opponents. In fact, in many cases you'll find these churches have ministries to reach out and assist the very people they supposedly "hate." No doubt there are some groups and people who truly hate other individual people, but I'm going to go out on a limb here and say they're doing it wrong.
Does this setup cause conflict? Yes. In the math example, the other "side" clearly believes they are right. In the conversations we have with others and the battles we must wage politically, the opposing "side" believes just as heartily as we they are indeed correct. Disagreement is inevitable. But let's get over the idea we have to hate each other, personally, in the process. "Remember who the real enemy is." For Christians, that enemy is satan. It is okay to hate him. But everyone else is called to a life with God, even if they don't know that, and we should remember it and treat them accordingly.