Extraction of calcium carbonate from liquid
The stuff you're seeing is solid calcium carbonate that most likely did not dissolve, or some other insoluble calcium compound that formed by reaction of calcium with whatever is in the apple juice.

You might want to try using sodium carbonate instead of calcium carbonate. Sodium compounds are usually more soluble than calcium compounds, and it's a more common food additive than calcium carbonate.

That said, I wouldn't recommend adding chemicals to your food without knowing exactly what you're doing. Some seemingly safe chemicals may have hazardous impurities and do not qualify as food-grade. If you're doing this because of health problems then you should consult with a doctor on better ways of action than pouring carbonates in your drinks.