Despite the Doctor being undoubtedly wise and well-versed in a variety of subjects, Doctor Who has never explicitly confirmed just what disciplines the Doctor holds a doctorate in. There's also some questions as to whether or not the degrees he holds are strictly honorary or what degrees he earned on his home planet of Gallifrey.
What little is known about the Doctor's academic life comes from two classic series episodes; "The Deadly Assassin" and "The Armageddon Factor." The former revealed that the Doctor attended the Prydonian Academy, though one Time Lord thought he had been expelled because of some sort of scandal before graduating. This accounting appears to be faulty, however, as another Time Lord named Drax recalled how the Doctor (then known as Theta Sigma) earned a doctorate as part of the graduating class of 1992. While this seems to confirm that the Doctor earned a doctoral degree of some kind on Gallifrey, it does nothing to reveal what he studied!
Whatever the Doctor originally studied, it seems unlikely he was a medical doctor. The First Doctor outright denied being a doctor of medicine when asked to treat an injured caveman in the first episode of Doctor Who, "An Unearthly Child." He continued to make this claim, denying he was a physician to Kublai Khan in "Marco Polo" and wishing he had earned his medical degree in "The Rescue." It is possible, however, that the Doctor might have been a medical doctor on Gallifrey (or at least studied medicine) but was unwilling to take a chance on treating biologically similar yet different species like humans. This would explain why the First Doctor gave the wrong advice when pressured into treating the victims of a fever-inducing virus in "The Ark," saying that all the patients must be wrapped up tightly and kept warm.
Despite this, the Doctor did eventually gain medical training in both human and alien medicine. The Second Doctor claimed to have studied medicine and earned a medical degree with Joseph Lester in 1888, when questioned on his credentials by his companion, Polly, in the episode "The Moonbase." By the time of his Sixth incarnation and the episode "Mindwarp," the Doctor was capable enough to oversee a cross-species brain transplant. In "Remembrance of the Daleks," the Seventh Doctor demonstrated the ability to diagnose human ailments by touching the patient's ear. In the new series, in the episode "The God Complex," the Eleventh Doctor claimed to have earned degrees in medicine and cheese-making.
Another possibility is that the Doctor might have been a lawyer and earned his doctorate in law. The Fourth Doctor kept a British barrister's wig in the pocket of his long coat and donned it while defending himself in the episode "The Stones of Blood." He also cited an obscure Gallifreyian law to delay his being put on trial for assassinating the President of Gallifrey in "The Deadly Assassin," by declaring his candidacy for the now vacant office of the President. (Reportedly, this law was introduced to thwart a previous President's attempts to cancel the election by having false charges brought against his sole opponent.) The Sixth Doctor elected to represent himself during "The Trial of a Time Lord," claiming that he had "been through several such inquiries before." It's also interesting that the villain of "The Trial of a Time Lord" was a corrupted future version of the Doctor known as The Valeyard; an archaic title for a Doctor of Law.
More often than not, however, the Doctor has presented himself as a scientist, albeit one a multi-disciplinary one. The Third Doctor claimed in "The Mutants" to be a doctor qualified in "practically everything" and the Fourth Doctor claimed in "Revenge of the Cybermen" to be "a doctor of many things." By the time of his Twelfth incarnation in the current series of Doctor Who, the Doctor had proven qualified enough to earn a professorship at St. Luke's University and the position of "Lecturer In Almost Everything." Given that, it seems safe to say that the Doctor has earned their title many times over, in the School of Hard Knocks if nowhere else.