There is nothing wrong with going to a non-Muslim doctor, if he is skilled and trustworthy
Question

I am suffering from a disease of the digestive system, and I am being treated by a Christian doctor. I have been treated by him for approximately two months, because I do not know of any other doctor. Then I found out about another doctor who is Muslim. Should I stop the course of treatment with the Christian doctor and go to the Muslim doctor, or should I complete the course of treatment with the Christian doctor?
Praise be to Allah.If this Christian doctor is honest, trustworthy and skilled, and you started treatment with him and saw progress and improvement from this treatment, then there is nothing wrong with continuing with him, and you do not have to leave him and go to the Muslim doctor.
A brilliant doctor – even if he is not Muslim – may save you a lot of effort, time and money, and Allah, may He be exalted, may decree healing at his hands.
The Muslims, in the past and in modern times, have always sought the help of skilled doctors – even if they are not Muslims.
Ibn al-Muqri’ narrated in his Mu‘jam (352) that al-Mubaarak ibn Sa‘eed said: When Sufyaan – meaning ath-Thawri – first began to follow the path of asceticism (zuhd), we thought that he was sick, so we took his urine in a bottle and went to a Christian doctor, who said: Your companion is not sick; there is nothing wrong with him except fear, and this is nothing but the urine of a monk.
Al-Mirdaawi said: I saw a Christian doctor come out of Imam Ahmad’s house, accompanied by a monk, and he said: He asked me to let him come with me to see Abu ‘Abdillah." (Siyar A‘laam an-Nubala’  11/211).
Ibn al-Qayyim (may Allah have mercy on him) said: The fact that the Prophet (blessings and peace of Allah be upon him) hired ‘Abdullah ibn Urayqit ad-Du’ali as a guide at the time of the Hijrah, even though he was a disbeliever, indicates that it is permissible to consult a disbeliever regarding medicine, treatments, writing, accounting, defects, and so on, and that does not come under the heading of appointing them to positions of authority that require a Muslim of good character. The fact that he is a disbeliever does not mean that he cannot be trusted with anything at all, for there is nothing more crucial than guiding people on a journey, especially in a journey as important as the Hijrah." (Badaa’i‘ al-Fawaa’id  3/725).
Ibn Muflih said, quoting from Shaykh al-Islam Ibn Taymiyah:
If the Jew or Christian is an expert in medicine and the individual trusts him, it is permissible for him to consult him, just as it is also permissible to entrust wealth to him and deal with him, as Allah, may He be exalted, says (interpretation of the meaning):
{And among the People of the Scripture is he who, if you entrust him with a great amount [of wealth], he will return it to you. And among them is he who, if you entrust him with a [single] silver coin, he will not return it to you} [Aal ‘Imraan 3:75].
If he can find a Muslim to give him medical treatment, and if he can find a Muslim to entrust his wealth to or to do business with, then he should not go to anyone else. But if he needs to entrust some of his wealth to a kitaabi (one of the People of the Book) or seek medical treatment from him, he may do that, and this does not come under the heading of taking Jews and Christians as allies, which is forbidden. And if he addresses him in a kind manner, that is good." (Al-Aadaab ash-Shar‘iyyah 3/76).
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
Is it permissible for a Muslim woman to be treated by a Christian woman?
He replied:
If she is trustworthy, there is nothing wrong with that. The evidence for that is the fact that when the Messenger (blessings and peace of Allah be upon him) travelled from Makkah to Madinah for the Hijrah, he employed a mushrik man called ‘Abdullah ibn Urayqit, from the tribe of Banu ad-Dayl, to act as his guide on the journey." (Liqa’ al-Baab al-Maftooh  2/56).
We ask Allah to grant you healing and well-being.