I know very little about the French system, but am surprised you could not find anything online. I did a little poking and found what seems like the answer; perhaps you or other users can correct me if this is overly simplistic.
[Si] Vous avez obtenu un diplôme aux Etats-Unis, et vous désirez rentrer au pays (France), dans ce cas : Vous devez contacter le Centre ENIC NARIC de Paris, en leur demandant une Attestation de Comparabilité du diplôme étranger avec un « équivalent » Français. Cette Attestation est délivrée au cas par cas, après étude suivant une grille d’analyse très précise établie par le Centre.
My rough translation:
[If] you have obtained a US degree and you desire to return to France, in this case: you should contact the ENIC NARIC Center in Paris, and ask for a Certification of Equivalence between the foreign degree and a French one. This Certification is granted on a case-by-case basis, after a very precise, point-by-point analysis as established by the Center.
So, it seems like if you finish your degree in the US and get this Statement of Equivalence, then you should be able to continue your studies in France as if you had obtained the equivalent French degree.
Note, however, that French doctors have to undergo substantial retraining before they can practice in the US. So, your decision is, at least, difficult to reverse. Given that you've never lived in France but suddenly want to move there "as soon as possible," I would urge you not to make any rash decisions.