Recognition of independence is not the same as a grant of independence.
The US, in particular, was not initially recognized as independent by anyone since nobody wanted to piss off Great Britain, until some of the First Nations (at the time recognized as fully sovereign) recognized the newly-formed United States as sovereign. (It's a precept of international law, by my understanding, that a country is sovereign when they are recognized as such by another sovereign country.) Only after that did other nations begin to recognize the US as independent.
(Which makes the mistreatment of the First Nations by the US as even more disgusting--they're why everyone recognizes you as independent, you ungrateful dipshits.)