The answer is "sort of". In some ways, you're asking the wrong question.
You need documentation of 6 months of completed training -- either for your international training, or for a medical license, or for whatever. Usually, for this type of purpose, all you need is a statement that "Lightbender worked at XYZ program from Date A to Date B and resigned in good standing". Usually, rather than a letter (which could be easily forged), any new program / licensing body will contact the program directly to confirm your dates and training. So, most PD's would not be vindictive and lie -- they will simply state that you completed training. Often this would be done by the GME office, who could confirm your dates.
When we talk about a "letter", we're usually talking about an LOR -- will your PD state that you did good work and that they support you in your quest for a new spot. That's a more complicated question.
So, in general, an employer does have a requirement to report what dates you worked, and whether your work was satisfactory. Your PD does not have any requirement to write an LOR (although I think that any decent PD shoulc write an honest letter of a resident's performance).
On the topic of your overall plan, I think you're taking a leap of faith -- and that can be a good thing. As you can see from the comments, you're going down the road less traveled. This could end with great career for you, or great regrets. But, you could have regrets not following your dream also. So, best of luck and I hope it works out.