Yes, this is correct. What you've done with the 's is omit the noun that's being modified, i.e.:
A society so vibrant and diverse as America's society is bound to dominate.
You can also omit the 's, which would imply that American is a society rather than having a society (which, to be honest, kind of mean the same thing).
However, the second sentence doesn't sound right to me:
A personality so mannered and educated as John's always has a bright future.
The problem here is that personality can mean "person" or "characteristic of a person," but not both at the same time. Now who/what can have a bright future? A person, not a characteristic. But if personality meant "person" here, you cannot use it with the possessive "John's," which only makes sense when applied to a characteristic. So changing "John's" to "John" would make the sentence correct, though it still sounds a bit eccentric to me (I'd just go with "somebody" at the beginning).