I know that you probably meant to express that Google is the opposite to a public (= governmental) entity, but "private company" is not the term to express that

I know that you probably meant to express that Google is the opposite to a public (= governmental) entity, but "private company" is not the term to express that. Google is a public company (= listed on the stock market for anyone to buy and sell), so the opposite of a private(ly owned) company.