"George Washington Smith"...you're kidding, right?There will be a hundred soldiers with this name.If you can pick your G. W. Smith off of the BLM website ( www.glorecords.blm.gov ), then you could order his files.
If he's 1812, he will have a service record or enlistment record, depending upon whether he was a volunteer or a "Regular Army" soldier.If he was a volunteer, his service record will not have his family or his place of birth.If he is a "Regular Army" soldier, his enlistment record MIGHT give his state of birth, but you can't count on it.
If he has a pension file, then you will get some personal information about him and might get the information you are looking for.No guarantees.
He will probably also have a bounty land application file and a surrendered warrant file.
The pension file and bounty land files would show where he was living at certain times, including where he enlisted.They may or may not show where he was born.Only rarely do they show who his parents were, unless his parents were living and gave affidavits to prove whatever he needed to prove to the government.