Can I Make Dua to a Christian Doctor?
Question

I do not like to go to a male doctor, and I prefer a female doctor. The only skilled female doctor I know is Christian. I feel comfortable with her treatment of me, and we talk together. When I talk to anyone, I always make Du`a (supplication) for them, saying, “May our Lord bless you, may our Lord honour you” and so on.  Is this supplication of mine permissible or not?
Praise be to Allah.Making Du`a (supplication) for a non-Muslim who is living under Muslim rule or whose people are at peace with the Muslims falls into two categories:

The first category is supplications having to do with the hereafter, such as praying that he/she be admitted to Paradise, or praying for forgiveness or mercy for him , or praying that he be ransomed from the Fire, or that he attain the intercession of our Prophet Muhammad (blessings and peace of Allah be upon him), and similar supplications.

It is not permissible to offer this type of supplication for him, because Allah, may He be Glorified and Exalted, has forbidden it, as He says (interpretation of the meaning):
{It is not for the Prophet and those who have believed to ask forgiveness for the polytheists, even if they were relatives, after it has become clear to them that they are companions of Hellfire} [At-Tawbah 9:113].
Abu Hurayrah (may Allah be pleased with him) narrated that the Messenger of Allah (blessings and peace of Allah be upon him) said: “I asked my Lord for permission to pray for forgiveness for my mother, but He did not give me permission.” (Muslim, 976)
An-Nawawi (may Allah have mercy on him) said in Al-Majmu‘ (5/120):
With regard to offering the funeral prayer for a disbeliever and praying for forgiveness for him, it is prohibited according to the text of the Quran and scholarly consensus.

The second category is supplications having to do with worldly matters, such as praying that he be given much wealth and many children, or praying that he be healed, or praying that he be prosperous and happy. One of the greatest and most important of such supplications is praying that he be guided .

This type of supplication is permissible; there is nothing wrong with it and it does not incur any sin, for several reasons:

There is no prohibition regarding it, and the basic principle is that things are permissible unless there is evidence to indicate that they are not allowed.
It is narrated in the Sunnah that it is permissible to return the greeting of Salam given by a disbeliever if he greets in clear words, and returning the greeting of Salam is a supplication for safety and well-being. It is also narrated in the Sunnah that it is permissible to do Ruqyah for a non-Muslim , and Ruqyah is a supplication for healing.  
It serves the interest of softening the heart of this disbeliever [towards Islam], which is a great and significant interest and one of the objectives of Shari`ah. The Prophet (blessings and peace of Allah be upon him) visited a Jewish boy who was sick; he called him to Islam and the boy became Muslim.
Similar supplications have been narrated from some of the early generations. Examples of that include the following:

It was narrated from ‘Uqbah ibn ‘Amir Al-Juhani (may Allah be pleased with him) that he passed by a man who looked like a Muslim in appearance. The man greeted him with Salam, so ‘Uqbah returned the greeting by saying: And [peace] be upon you and the mercy of Allah and His blessings. The slave [of ‘Uqbah] said to him: He is a Christian. ‘Uqbah got up and followed the man until he caught up with him, and said to him: The mercy of Allah and His blessings are for the believers, but may Allah give you a long life and give you much wealth and many children. (Narrated by Al-Bukhari in Al-Adab Al-Mufrad, 1/380).
It was narrated from Al-Hasan Al-Basri (may Allah have mercy on him) that he said: If you offer condolences to a Dhimmi [A non-Muslim living under Muslim rule], then say: May nothing befall you except that which is good. (Narrated by Ibn Al-Qayyim in Ahkam Ahl Adh-Dhimmah (1/438); he also narrated a number of similar reports).

The jurists (may Allah have mercy on them) also regarded this type of supplication as being permissible. There follow some of their statements concerning that:

It says in Kashshaf Al-Qina‘ (3/130), by Al-Bahuti Al-Hanbali (may Allah have mercy on him):
“It is permissible to say to him [a disbeliever]: Welcome, how are you? and similar phrases, such as, How are you doing? It is also permissible for a Muslim to say to a Dhimmi: May Allah bless you, may Allah guide you – meaning: to Islam. Ibrahim Al-Harbi (may Allah have mercy on him) said to Ahmad: Can we say to him: May Allah bless you? He (may Allah have mercy on him) said: Yes, meaning: with Islam.”
It says in Hashiyat Nihayat Al-Muhtaj (1/533) and Hashiyat Tuhfat Al-Muhtaj (2/88), which are Shafi`i books: “It is permissible to offer supplication for a disbeliever, asking that he be granted good health and guidance." 
Al-Munawi (may Allah have mercy on him) said in Fayd Al-Qadir  (1/345):
“It is also permissible to offer supplication for a disbeliever, asking that he be granted guidance, good health and well-being, but not forgiveness.”
Based on that, there is nothing wrong with you making Du`a for this Christian doctor by saying the words mentioned in your question: May Allah bless you, may Allah honour you – intending thereby to ask Allah, may He be Exalted, to bless her and honour her with Islam.
Imam Ahmad ibn Hanbal (may Allah have mercy on him) was asked about a Muslim man who says to a Christian man: May Allah bless you. He said: Yes [that is permissible]; he may say: May Allah bless you – meaning with Islam." (Al-Adab Ash-Shar`iyyah by Ibn Muflih  1/369).
For more details, please see the following answers: 128862 and 13611 .
And Allah knows best.