AFAIK, an MRI does not necessarily needs to be prescribed, you could in principle simply make an appointment with an imagery lab. Usually, patients would have a referral letter by their general practitioner or another physician, which would indicate what needs to be done and why but that’s not really a “prescription” like the ones you need to get restricted medicines. Having a letter from your physician in the US could still be helpful but since it’s not formally a prescription, recognition should not be an issue (but you would obviously need to find a radiology specialist who is able to read English).
The lab will probably ask for this letter (along with a number of other things) and it might be difficult to get an appointment without it but it is by no means mandatory. Because of this, you could probably even get money back from the French state insurance system if you are covered (unlike medicines prescribed from non-EU physicians, which are not covered). As user13619 commented below, you might however get a 10% penalty since you would be outside the parcours de soin.
Anecdotally, I heard there is some waiting time for MRI so it might be difficult to fit it in your holiday if you haven’t made an appointment beforehand but it’s certainly worth a try.
Credit:stackoverflow.com‘