An authorization send to another country can it be notarized here in the United States?

In many cases it can but not every country will accept it. What country are you dealing with and what type of document are you going to be signing?

The notary process depends on what you are having, and why,
The law around that specific legal aspect.

Is it a deed to sell a house, a will, a car sale, something for family court ?

Remember a notary is merely a very legal method of getting a witness that you signed something, it does not make it any more "legal"

So you will have to meet the requirements for your signing in the area where the document is going to be used and for that purpose.

For example, if you need a document for family court in GA signed, then the notary or witness rules for that court has to be followed