Thank you for your question. I think you were considering going to Ukraine to sue your ex-husband.
However, I do not think that is necessary. According to EU Regulations, and to the Hague Convention on Parental Responsibility, all questions of child custody, separately from divorce, may be tried by the courts of the state where the child is "habitually resident" which, going by your question, are French courts.
You would need to go to your local Tribunal de Premiere Instance and sue your ex-husband for the custody, and he would be summonsed in the Ukraine, but you do not need to go to the Ukraine yourself in any way. The further advantage is that such a ruling can be easily recognized in the US, and is automatically valid in France.
The fact that your divorce ruling does not specify the child is in fact good, because your husband cannot claim that the Ukrainian court already decided on the question of custody. Since that question is undecided, you may apply to the French courts without a bar to your action.
I strongly suggest taking a local French attorney specializing in "private international law" (droit international prive) which is the branch dealing with international family matters. Legal help is available in France, for any resident, according to EU Directive and international agreements, so you may be able to obtain the attorney for free (see here: https://www.service-public.fr/particuliers/vosdroits/F18074).
I hope my answer is useful and look forward to your rating, which is essential to my activity.
Dr I L Vlad