Unfortunately, the French regions (France and French-speaking Switzerland) are a virtual impossibility without EU citizenship, no matter how well you can speak the language. According to chapelier, a French medical student in Romania who posts on VMD, as a non-EU citizen with a non-EU diploma, you could potentially apply to train in France as a GP only... and you would have no practice rights in France thereafter. Even that training is not guaranteed by any means.
The most feasible option right now in W-EU is probably in Germany, although it is getting tougher there and it is definitely more difficult without an EU diploma. Immigration is likely an issue, so you would need to speak the language with a very good level of fluency and be seen as someone who could "integrate" well into German society. Germany has an excellent medical system and will not tolerate poor language skills or shoddy medical work -- I mention this not because I suspect you would be a bad hospital employee, but rather because a few German hospitals accepted some bad foreign doctors in the past years with tragic results (i.e. dead patients, etc.) and they are really cracking down on foreign physician employment in many areas. If you want to try for Germany, I would recommend doing a Google search German medical recruiters... there are hundreds of them. They will be able to tell you what you should do to apply for an Internal Medicine position in Germany, and what your chances are. On the good side, IM is a specialty that seems to have a lot of openings there at the moment.
Otherwise, you would likely be looking at E-EU (Bulgaria, Romania, perhaps Poland, etc.) as the most likely locations for your IM training. This would mean that you would have to pay for your residency training, rather than being paid for your work, which would be a dramatic shift of financial fortunes. Still, depending on where you want to work, it could be a potential path to your goal. We have an Indian physician working as a resident in cardiology in Sofia, and he seems to be doing fine... but be aware that there is a significant drop-off between the health care systems of W-EU and E-EU.
"To array a man's will against his sickness is the supreme art of medicine."
- Henry Ward Beecher