South Africa, Namibia, and India may be good options. I have had a lot of experience with clinics/hospitals in South Africa and would highly recommend this location. However, keep in mind that while English may be the official language in these areas among doctors etc, it isn't always spoken that often around the hospital, as the patient base is usually not English speaking.
I would recommend that you do look into projects relating to public health, as this is an excellent way to be exposed to issues very pertinent to the practice of clinical medicine, but you may also contribute in a meaningful way without a medical degree.
There are many ways to find international internships or research experiences. Many people I know went to international health conferences, and talked to visiting researchers there whose work seemed interesting. Others worked directly with faculty from their university who had projects and connections abroad. In these cases prior research experience and strong letters of recommendation may be needed.
A simple google search will reveal many NGOs who you can directly email and ask about volunteering. Be prepared with a project proposal and letters of reference in most cases. International work is also notorious for being underfunded, so paying your own way is common practice, though living expenses or homestays can often be arranged.