The reaction you write, in aqueous solution, would happen with the ions produced by dissociation of the salts, not with the parent compounds, and would involve not one but two acid-base reactions (with carbonate ion, and then bicarbonate) and finally decomposition of carbonic acid to $\ce{CO2}$.
So while the complete reaction is
$$ \ce{CaCO3 + H2SO4 -> CaSO4 + CO2 + H2O}$$
there are at least the following steps involved (all species aqueous except solid $ \ce{CaCO3}$ and $\ce{CaSO4}$)$^1$:
$$ \ce{CaCO3 -> Ca^{2+} + CO3^{2-}} \\ \ce{H2SO4 -> 2H+ + SO4^{2-}}\\ \ce{Ca^{2+} + SO4^{2-} -> CaSO4 }\\\ce{ CO3^{2-} + H+ -> HCO3- }\\\ce{ HCO3- + H+ -> H2CO3 }\\ \ce{H2CO3 -> CO2 + H2O}$$
You can then see that bicarbonate $\ce{ HCO3-}$ can act as either acid or base (in the Brønsted-Lowry sense) whereas $\ce{ CO3^{2-}}$ acts as a base (proton acceptor).
Note also, regarding the Lewis definition, that Brønsted-Lowry acid/bases form a subset of Lewis acids/bases.
- It might be worth noting that $\ce{CaSO4}$ is slightly soluble in water (on the order of g per L at RT), and that carbonic acid is not stable in the presence of water at RT.