This about it this way. Calcium carbonate dissolves to some small extent, producing CO32-:
CaCO3 <-> Ca2+ + CO32-
At the same time carbon dioxide dissolves in water, producing carbonic acid. So now you have carbonic acid and CO32- in solution:
H2CO3 + CO32- <-> 2HCO3-
You may also write carbonic dissociation first stage
H2CO3 <-> H+ + HCO3-
and
H+ + CO32- <-> HCO3-
which yields exacltly teh same result.
But as I wrote earlier - no matter what is the system involved, when you add acid, pH goes down, when you add base, pH goes up.