Nothing, except some airline rules requiring a medical examination and the agreement of the captain to allow pregnant women in the 7th, 8th, and 9th months to embark on transcontinental flights, to fly back to the US.
If he finds out he is the father of the child, even a long time after the baby is born, he can recognize it as his, provided no other man has done some before him. If there is an argument on the matter either with the mother or another putative father, a court will rule on it. The father may ask that genetic tests be used.
If the child is recognized by his father before he or she leaves France and he asserts that he wants to either share custody or have visitation rights, and the mother takes the child abroad, she may face prosecution for child abduction and, due to international treaties, may have to take him/her back to France.
If a father has a strong case, he may win sole custody, or at the minimum shared custody even if most children are left with their mothers.
DISCLAIMER: Answers from Experts on JustAnswer are not substitutes for the advice of an attorney. JustAnswer is a public forum and questions and responses are not private or confidential or protected by the attorney-client privilege. The Expert above is not your attorney, and the response above is not legal advice. You should not read this response to propose specific action or address specific circumstances, but only to give you a sense of general principles of law that might affect the situation you describe. Application of these general principles to particular circumstances must be done by a lawyer who has spoken with you in confidence, learned all relevant information, and explored various options. Before acting on these general principles, you should hire a lawyer licensed to practice law in the jurisdiction to which your question pertains.
The responses above are from individual Experts, not JustAnswer. The site and services are provided “as is”. To view the verified credential of an Expert, click on the “Verified” symbol in the Expert’s profile. This site is not for emergency questions which should be directed immediately by telephone or in-person to qualified professionals. Please carefully read the Terms of Service (last updated February 8, 2012).