Does a father have a right to have his last name as his child's last name on the child's Birth Certificate in the State of Georgia?

Nope, mother names the child. If you are married, and mother not able to name because of health, you can name, but if not married, no rights what so ever And if they are not married, if he is not listed on the Birth certificate, he has not rights to child until he goes to court and proves he is the father and requests a level of custody

Here in the States it is strictly up to the mother of the child as to what last name goes on the birth certificate. I had this discussion the other day with the "birth certificate lady" who works in the hospital where I work. There are legal ramifications that are either easy to change the name later or virtually impossible to change.

When the child is born, the birth certificate lady will come talk to you and can answer all of your questions.

You failed to mention the most important data, so...

Generally that is standard when :

1 - the father is married to the mother of the child.
2 - the father accepts the child as his own.

If you are not married to the mother, it is normally up to the mother to decide on the family name of the child, and than her decision often depends on the secondary arrangements made, like child support arrangements.
You can make such arrangements on a friendly basis, and follow that up - to make it official - by a court procedure.

I do not know if Georgia is an exception to that, though...

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