It's true that acids dissociate in water to their respective ions. For example; acetic acid forms acetate ion and hydronium ion in water.
CH3COOH + H2O <---> CH3COO- + H3O+
CO3-2 ion cannot dissociate further as it is not an acid but an ion. It is formed by dissociation of carbonic acid which is a weak organic acid.
H2CO3 <------> 2H+ + CO3-2