Which one is correct: "Don't wait for us. We aren't swimming/aren't going to swim today, we're tired.
Both are grammatically correct but, in a realistic situation, neither is natural. The reason for this is that, for most people, they have to go somewhere- for example, a beach or a swimming pool, in order to swim. So most English speakers would think of the activity as go/going swimming, or go/going for a swim. In the same way, one might go to the cinema or go out for a meal. The going is an important part of the activity.
Putting that idea into your sentence, you get:
Don't wait for us: we aren't going swimming today.
Don't wait for us: we aren't going to go swimming today.
Don't wait for us: we aren't going for a swim today.
Don't wait for us: we aren't going to go for a swim today.
The don't wait for us part of the sentence lends weight to the idea that the group of people in this particular situation would have to go somewhere to go swimming.