I personally hate the concepts of fate and destiny, especially when given as the reason as to why two people are together. It is my belief that "it's destiny" cheapens the real effort a relationship needs, and makes it look like the eventual conclusion is out of the hands of the people involved. Why is fate, the idea that everything is predestined to happen anyway, supposed to be such a romantic concept?

Roy & Hawkeye, for me, represent the opposite of that which is foreordained. Much more important than how two people meet is how they stay together. They may have met again by chance, but coincidence is not the same as fate — especially when said "chance" meeting wasn't that unplanned, really.

Roy was present when Hawkeye's dad passed away. The young man was his alchemy apprentice, but there was nothing he and his teacher's daughter could do to save his life. Professor Hawkeye was an absorved man, to the point where his own child was scared of his obsession. Still, he regretted not sharing his full knowledge to Roy and thus left Hawkeye in his care, in more ways than one.

Hawkeye was nearly an adult by then, capable of taking care of herself without Roy's help, but he instead gave her something that would stay with her for the rest of her life and which she'd learn to cherish and protect.

Roy: [...] if I could become one of this country's foundation stones and be able to protect everyone with these hands, I think I'll be happy. [v15 ch060]

It was still too early for them to discover just how naive that dream was, but Hawkeye found Roy's words sincere and so she decided to trust him with her father's knowledge — a research carved on her own back's skin.

They parted ways for some years and then met again in the battlefield — in the middle of the Ishval War, a civil uprising turned massacre. Hawkeye joined the military after feeling inspired by Roy's dream, but soon found herself broken and cursing that which had brought her there. Roy wasn't much better, the guilt of betraying the trust she had placed in him eating his insides. Both turned into killers, a greater goal would soon appear in front of the two.

After the war ended and Roy had lost all faith in the military institution of his country, he decided to stand up and try to change it all — from the inside. Hawkeye had decided to make a martyr out of herself by staying in the military, but Roy had a new dream for her.

Roy: I'm thinking of recommending you as my aide. I want you to protect my back. Do you understand? To entrust my back to you means that you can shoot me from behind any time. If I step off the path, shoot and kill me with those hands. You are qualified to do that. Will you follow me?
Hawkeye: Understood. If that is your wish, then even to hell. [v15 ch061]

They would keep their lives in the military that had caused them so much pain, but with a new objective in sight. Not only that, Roy was also placing the ultimate success of their mission — as well as his own soul — in her hands. Roy had already briefly lost her trust once before, and now he was determined to never let go of it again.

East City was Roy & Hawkeye's healing place after the war. The traumatic moment in their lives that was Ishval gave them the push to dedicate their lives to a just cause, but I think the time spent before and after that moment is what made them who they are. They've spent such a big portion of their lives together — not because they're forced to, but because they seek the company — that it's unavoidable they influenced each other.

Perhaps it was sheer luck what made them meet, but it was their will what made them stay by each other's side. They've grown as people together, each one of them contributing to the other's development. They've become better by staying together and, perhaps most importantly, they are happy staying together.