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The point of public domain code is that you can do anything with it that you want. You could even release it under the GPL.

This is a derivative so yes, you can (just like you can release derivatives of MIT-licensed code[1] under a license that doesn't let the recipient receive the source code, or conversely under a copyleft license).

[1] "derivatives" include compiled binaries

Genuine question - what do you think "public domain" means?

The problem is more that "public domain" is a concept that is not universally accepted in a legal sense, examples are Europe and in particular Germany. Case in point, Sqlite had to come up with a license agreement to make it usable for companies in those countries.

Public domain exists everywhere, but the concept of releasing something into the public domain by choice doesn't, so in those jurisdictions something only officially enters the public domain some time after either the author's death or publication.

The mental model of GPL cancer has infected the concept of public domain