Giving her up doesn’t mean he can’t love her. He just had to learn that he couldn’t always put her above everything else. That lesson hit home in the catacombs when he realized that by choosing his attachment for Katara, he was actually putting her in danger because he wasn’t able to protect her from Azula, Zuko and the Dai Li agents who vastly outnumbered them. When he realized that there was no way for them to win, he chose to let go of Katara in order to give them a chance of surviving the fight. So really, it was his love for Katara that motivated him to let go.