“The operative word here is “can”. In a rational society, the word would be “shall”, but we have once again been blessed by the courts with another example of their inherent stupidity when they ruled that State officials don’t have to do their job. They can if they want to, but they cannot be compelled to do their job if they don’t want to do so.”
The reason that no criminal court procedure has gone forward regarding Barack Obama’s birth certificate is because the original, hard copy, vault edition has never been seen publicly. People have reacted to a third or fourth generation image that was scanned to pdf. That is an image, not a vital record of birth. The original remains in the office safe of the Hawaii Registrar where it has resided since 2011. The original can be made available for inspection via court order.
The mechanism of proving eligibility need not be a birth certificate, but that is the normal method nowadays for doing such. So far as there is a law, it's Federal, and explicitly mandated by the body of the US Constitution, so it is higher than just ordinary statutory law such as something Congress would create.
The original can be made available for inspection via court order.
Except the public has no standing to bring a case, and those who did have standing, had no interest in bringing a case.