It doesn't matter that Aang is the Avatar.
It doesn't matter that Aang is possibly his best friend.
It doesn't even matter that Aang is probably too naive to even think anything remotely similar to that.
Aang, no, no-one, is touching his sister. Katara is the only thing that has been consistent. She's always been there for him, always tried to heal him. Nobody is taking Katara away from him.
Because when she leaves, what will he protect? What will he give his all to save, and teach, and shield? Who will he talk to about Suki, and Yue? Who will he tease?
Aang is a good kid. Aang is kind, silly, serious, and strong enough to protect her. Aang is his best male friend. But Aang is not good enough. No-one is good enough for Katara.
And if Katara hates him forever for protecting her against the not-quite-worthy, then so be it.