The answer will depend on the laws of your country.
In general, the rights of the sender for privacy etc. have to be weighed against the public's need to know.
Private communication is private and in most cases must not be published, because otherwise the personality rights of the sender are being violated. But if that letter contains information that would help solve or prevent a crime, the need of the public to know of that crime outweighs the rights of the sender.
Similarly, commercial letters are to be kept confidential if they contain information that the sender has a right to keep confidential, but if the letter contains information that is of importance to the public (e.g. unlawful business practices), that information might outweigh the right for confidentiality.
Commonly, only a court of law can decide if publishing (parts of) a letter was lawful.
If you want to avoid breaking the law, you can anonymize the information in the letter, so that there is no possibility for anyone to identify the sender (and in this case, receiver) of the letter. That is, even your family who know of your visit to that doctor must be unable to conclude that it was this or that doctor who wrote (and who read and reacted or failed to react appropriately) to the letter! If you cannot completely anonymize the quote, and the quote would throw an unfavourable light on any of the persons involved, I would consult a laywer before publication.
If the quote is harmless and would not be considered slander by anyone, even the doctors themselves, an anonymized publication seems unproblematic to me.