What is wrong with my equation?

Actually, carbonic acid is formed. However, carbonic acid is very unstable and thus it decomposes to carbon dioxide and water immediately.

Na₂CO₃ + 2HNO₃ → 2NaNO₃ + [H₂CO₃] → 2NaNO₃ + CO₂ + H₂O
________________________ (unstable)

Remember the general equation
Acid + Carbonate = Salt + Water + Carbon Dioxide.

The acid is Nitric Acid (HNO3)
The Carbonate is Sodium Carbonate (Na2CO3)
The Salt is Sodium Nitrate (NaNO3)
H2CO3 is Carbonic ACID , & being and acid it will react with the (excess/unreacted) carbonate ultimately liberating water and Carbon Dioxide.

Remember the General Equations for Acid reactions.
Acid + Alkali = Salt + Water
Acid + Base = Salt + Water
Acid + Carbonate = Salt + Water + Carbon dioxide (Given)
Acid + Metal = Salt + Hydrogen

NB An Alkali is a soluble Base. Not all bases are soluble.

There is nothing really "wrong" with it, apart from that isn't what ends up happening. Total carbonate content (all of the carbonate species combined, H2CO3+HCO3-+CO32-) will increase when you dissolve the sodium carbonate in acid.

Which of the three species will dominate will depend on the pH of the solution, but since you are adding a strong acid, the pH will rapidly decrease and bring the solution into the H2CO3-dominant field. Once you get there, there is no further you can go. And this is what you correctly predict.

However, there is a maximum quantity of total dissolved carbonate that can exist in solution, and that maximum is directly related to the pressure of CO2 gas in contact with the solution. If you acidify into the H2CO3-dominant pH field (which you clearly will unless you use only a trivial amount of HNO3), then you also have to consider the equilibrium exchange of CO2 gas with water via CO2+H2O=H2CO3. That is, once you migrate into the H2CO3-dominant field (low pH conditions) any further addition of dissolved carbonate will increase H2CO3, and that MUST dissociate in an "effort" to bring pCO2 into equilibrium with it. Basically, you will end up with too much H2CO3 for the amount of CO2 in the adjacent gas phase. Thus, there is degassing, CO2 is evolved, released.

That is, your reaction does occur, but the system does not stop there, unless you have a very narrow set of conditions. You can only add a relatively tiny amount of total dissolved CO2 to the system before it will supersaturate. Basically, water does not hold a lot of dissolved CO2 except when there is a very high pressure of CO2 gas in contact. The atmosphere doesn't have that kind of pCO2, so the solution will release some in order to make it get higher (which it never will in an open system, but in a closed system, yes, that would eventually be reached, and the solid carbonate would no longer dissolve once that was achieved).

This is one of those things about chemistry as a practical thing. Many reactions are real and will happen, but are not the ultimate (final) system reaction. Your reaction does occur, but dissolution would stop if there was no degassing. Clearly, that does not happen in most "normal" systems. Degassing will occur, must occur, at least until pCO2 is at the amount required for equilibrium with the solution.

So, your reaction is not "wrong", it simply is not what happens in total.