These days I believe HIPAA forbids such a release. When I worked at a health organization, we were not supposed to leave patient records face-up at our desks, if we were going to be away from our desks.
And even I/T people were not allowed in the records room where the main body of physical files were located. There were no computers in that room anyways that I recall being mentioned.
IOW, Obama would have to provide an (probably notarized) affidavit before such things could happen anywhere. Hawaii / hospital / etc.
I have always been able to get certified copies of death certificates--mine, my wife's, my children's, my grandfather's, even my great grandfather's. All arrived with an official seal on them.
I haven't tried to get copies of the birth certificates of George W. Bush, John McCain, or any other President, but it's my understanding that the first tow have released all such records to public scrutiny. I wonder why Obama hasn't.