I agree with both points OBP made above. Just to expand: a "communication" issue may have very little or nothing to do with specific difficulties with the English language or grammar. Rather, it is about the quality of one's communication with patients, families, co-workers, and colleagues. This issue transcends accents and Queen's English. Many residents struggle with the quality of their communication whether they are foreign born or from the US. As OBP alluded to, there are many fine physicians who speak in heavily accented English, who ensure the quality of their communication, nonetheless. I'm sure it is emotionally difficult to cope with this non-renewal, but use this opportunity to grow and learn what your program felt inhibited your ability to be an effective communicator. If you are open and willing, this might be something remediable and will serve you better when you reach the next step in your career progression.
Question: are you leaving in good standing? I always interpret a non-renewal of a contract as leaving in good standing (vs. being fired from the residency after failed remediation), but I'm sure the nuances of that may exceed my ken. Obviously, leaving with the program not willing to state that you would be suitable for further training somewhere else would be a difficult hurdle to surmount. But, if you are leaving with them willing to say: "yes, this person is suitable for further training elsewhere" you are in a better position. But understand, further training in pediatrics may not be in the cards for you. As part of your meeting you and your PD should sit and talk about what fields he/she feels you are suitable for. Sometimes they might think family medicine might be more appropriate, but (from what I've gathered from some of the wiser posters on SDN here) more often it is a field like psychiatry, pathology, or PM&R-ones with their own clinical challenges, but challenges much different than you face in pediatrics or internal medicine.
Just a few thoughts. Good luck to you.