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.