This is FUD designed to discredit intuitionistic logic.
An instance of LEM is not "a logical error"; it is merely a claim which requires justification. For the claim at hand -- i.e. some natural number G is a member of a finite, constructed set, or it is not -- the corresponding LEM instance is perfectly true [and proven]. We say that the claim is decidable. So (given that we have a proof the claim is decidable) there would be no problem in intuitionistic logic with this part of Euclid's proof even if it took the form of a proof by contradiction (which of course it doesn't).
To make a more general rebuttal of the FUD, proofs by contradiction are not automatically invalid in intuitionistic logic; they just require one extra piece of evidence.
Furthermore the claim that whether something is a proof by contradiction depends on one's position on the logical validity of LEM is ridiculous on its face. There is a fact of the matter as to whether any given proof is by contradiction. There may be disagreements as to the definition of "proof by contradiction", but the definition which includes proof of negation is completely useless. In mathematics we usually treat useless definitions as in some sense "false" in order to facilitate communication - e.g., 1 could be prime depending on definitions, but that would be useless so we facilitate communication by treating "1 is prime" as false. Likewise here we should treat as false the claim that the quoted section of Euclid's proof is by contradiction.
Source for claims about LEM/proof by contradiction: https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...