Well... The sparkling will be really intense and short-lived, as all that carbonic acid reacts with the bases on the milk. Then it will be completely flat.
Also, from the taste of old tooth-paste, I imagine CaCO3 tastes like spoiled sand would if sand was capable of spoiling.
I don't know if this is a true fact, nor will I be baking anytime soon. But this information will sit in my brain for years until one day someone is baking and I'll either look like a genius or ruin a good cake.
i used to do this as a kid. plop one of these https://en.wikipedia.org/wiki/Effervescent_tablet (but with fruit flavoring and sugar) in a glass of water, let it fizz a bit, then pour a bit of milk on top. the milk would instantly curdle and stay on top in a layer. it tasted bad. curdled milk with sugar and artificial lemon flavoring. i usually just poured it out after confirming that it still tastes bad.
i have no idea why i did this, it just looked fun.
That's not how they make fizzy drinks, they directly add the CO2. The acid thing is a myth, you'd just get very sour and weird tasting water with the slightest bit of gas since that process is too slow to be viable for this use.