Short answer
Yes. You may be treated by any competent, trustworthy doctor, Muslim or not; the Prophet ﷺ hired a non-Muslim guide for the hijrah because of his skill. The one Hanafi nuance is for religious concessions such as not fasting: the texts ask for a Muslim doctor's advice or your own experience, though many muftis today accept a reliable non-Muslim specialist.
Treatment by a skilled non-Muslim doctor is permitted. To take a religious concession on medical advice (breaking a fast, tayammum, praying sitting), the Hanafi texts require the word of a competent Muslim doctor who is not openly sinful, or one's own experience; many contemporary Hanafi muftis also accept a trustworthy non-Muslim specialist where Muslim doctors are not available.
The four schools agree on this.
Explanation
Medicine is a skill, and Islam judges a skill by competence and honesty. During the hijrah, the Prophet ﷺ and Abu Bakr hired a guide from Banu al-Dil who was still on the religion of Quraysh, because he was an expert in the desert routes, and they trusted him with their camels and their lives. Imam al-Bukhari placed this under a chapter on hiring non-Muslims when needed, and scholars took from it that you may rely on a non-Muslim expert in worldly matters.
In the US, UK, Canada and the Gulf, much of your care will come from non-Muslim doctors and nurses, and there is nothing wrong with that. Choose doctors for their skill, ask for a same-sex clinician where it matters to you and is available (see can a woman see a male doctor), and tell them about your needs, such as fasting, halal ingredients or modesty.
Religious concessions. When you rely on a doctor's advice to take a concession, such as not fasting, making tayammum instead of wudu, or praying sitting, the classical Hanafi texts say the doctor should be a competent Muslim who is not openly sinful, or you should rely on your own clear experience or obvious signs, because the advice changes a religious duty. Many muftis in Western countries accept the advice of a trustworthy non-Muslim specialist, especially when no Muslim doctor is available and the advice is ordinary medical opinion.
Never refuse treatment or ignore serious advice because of the doctor's religion. For fasting while ill, see can you skip fasting if you are sick.
Ask a mufti about your own case. This answer explains the general ruling. When the details of your situation matter, as they do here, ask a qualified mufti and give them the full facts. Where to ask a scholar.
The evidence
Sahih al-Bukhari 2263
Sahih... واستأجر النبي صلى الله عليه وسلم وأبو بكر رجلا من بني الديل ...
"The Prophet and Abu Bakr hired a man from Banu al-Dil (as their guide)."
Relying on a skilled non-Muslim in a worldly matter is allowed.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Sawm, on the excuses that permit not fasting. Requires the word of a skilled Muslim doctor who is not openly sinful, or one's own experience.
- Umdat al-Qari (al-Ayni), Kitab al-Ijarah, chapter on hiring non-Muslims when needed. A Hanafi explanation of relying on a non-Muslim's expertise.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.