I'm not sure I precisely know what you mean. Visa or no visa, if a child is born in a US hospital, they receive a birth certificate. So long as that birth certificate isn't for a child born to a foreign diplomat, the federal government - due do Ark - must recognize the birth-right citizenship of that child.
But, directly to your point, it's not likely, for two reasons. First, births are recorded by county bureaus of vital statistics (or some other such similar names). They, as county agencies, are governed by state law, not federal law. In fact, because of federalism, the federal government is precluded from mandating what an actual birth certificate has to look like, or what information it must include. The CDC suggests guidelines, they don't mandate guideline. To put is simply, it's the state that decides who gets birth certificates, not the federal government. Of course, for the purposes of citizenship, the federal government recognizes virtually any birth certificate that reflects a US birth, except in cases where the child is the son/daughter of a foreign diplomat.
Second, as Will points out, whatever the original intent of the 14th Amendment was, it's practical interpretation by SCOTUS (as seen in the Ark v. US case) is clear. The Supreme Court reversing itself in this regard seems HIGHLY unlikely. Ergo, the only legally practical solution resides in a constitutional amendment. Of course, that doesn't seemly very likely, at all, given the political climate in the country.
People keep mischaracterizing Wong Kim Ark. The ruling in that case only goes so far as aliens legally resident. Statements in that case and other cases suggesting birthright citizenship for children of aliens present illegally are mere dicta. The distinction is significant.
To clarify my earlier post:
Case 1: A child born to a parent who is visiting the US on a valid Visa, is a citizen of its parent’s country.
Case 2: A child born to a parent who is visiting the US without a valid Visa, is a citizen of the USA.
The statutes covering Visa’s could be rewritten, to grant the rights of “case 1” to the situation of “case 2”.
A simple reclassification of travelers to the US, to extend (certain aspects of) the same treatment provided to Visa holders to “non-Visa holders” would do the trick. No constitutional amendment would be required.
Well yes if they fit the same conditions as Wong Kim Ark did. Namely child of legal residents. But the children of illegals do not fit that mold, their parents did not subject themselves to the jurisdiction of the US.