Not likely no. Had Aang not frozen himself; he would have likely died in the Icy water. That would have been around less than 100 years before Katara would have been born. So, it would be almost impossible for Katara or Yue (another popular character theorized to have been "destined" to be Avatar had Aang not survived) to become the Waterborn Avatar.

@Zealous Gamer I meant if Aang had not run away and lived a full life as the Avatar.

He would have likely died either too late or too early for Katara to be born the Avatar. Remember, he lived around less than 100 years before Katara would be born. The timing would have to be exactly right for hher to be born around that time frame. Add to that there would be no guarantee that Katara would become the next Avatar. Just because the Korra came from the Southern Water Tribe; doesn't mean the next Avatar after Aang would have always been destined to be from the Southern Water Tribe.

I always thought that raava choose the next avatar after the current avatar died and then there was a pause for raava to choose the right person.

But if someone was supposed to be the avatar and fate was already decided ore something i think it would have been Yue earlier then Katara, she needed a spirit to live

Do you realize 153 AG Aang died and Korra was born so that means Katara would have to be born 153 AG

I don't think the timing would be right