No, we don't celebrate Children's Day in the United States. Not really.
In reality there are literally hundreds of special "days" and I am sure that there is a Children's Day. Wikipedia mentions the International Children's Day, the second Sunday in June. But you will not see much in the way of observance.
Mothers' Day, the second Sunday in May, is taken seriously. Many people do something about it. They send greeting cards. They take mothers out to restaurants for dinner, and give mothers gifts.
Fathers' Day, the third Sunday of June, is much less important. It's mostly a "commercial" observance. Stores will urge people to buy Dad a gift, and people may--but it's usually something small, perhaps a belt or a necktie.
If you ask people "when is Children's Day," I don't think many people will know, and they might say "Is there one?"