The simple answer is… both (see Who are the Morbius Doctors? for the how).

While it’s unlikely to be addressed by more than a one-liner in the new series, it should be noted that the TV series and expanded continuity actually contradict each other quite clearly on the matter of the Doctor.

In the TV series natural reproduction was clearly the de facto method of reproduction for the Doctor’s people. Susan was his granddaughter, and the first Doctor’s production team planned to introduce an evil son for the Doctor to do battle against. Later, the fourth Doctor and Romana both referenced their childhoods, and in The Creature from the Pit the Doctor reveals that he was born “under the sign of the Crossed Computers, the symbol of the Maternity Service on Gallifrey”.

The seventh Doctor also makes mention of family, and this trend has been continued in the new series, where the ninth Doctor reveals that his family are all dead and the Tenth tells us that he was a father once (which could, of course, be a reference to Miranda, the eighth Doctor's adopted -- and possibly biological -- daughter in the novel Father Time).

In the books, Lungbarrow redefined Gallifreyan reproduction once and for all, revealing that the people of Gallifrey were cursed with sterility, which in turn led to the creation of genetic Looms and to Rassilon unlocking the secret of regeneration. Based on plans originally drawn up by Marc Platt and Andrew Cartmel for the never-made season 27, the concept of the Looms was a powerful. Mythic, ironic, and fitting for the flawed history of a stagnant race like the Time Lords.

Bought into by much of fandom at a time when books were seen as the only possible future for the Doctor’s continuing adventures, the Looms inspired some of the ideas that drove Lawrence Miles in his novels Alien Bodies and Interference, and in his Faction Paradox spin-offs, where interbreeding with humans was seen as a legitimate means of attempting to break the curse.

Examining the evidence, the Doctor could be a father, a grandfather, a brother and even a husband without having been born. Even without a birth process, there could be Looms lined up along the walls of the Maternity Service, ready to churn out newly loomed Gallifreyans for eager-to-be-assigned parents.

But the Doctor has memories of being both loomed, and born. He has a belly-button, and has done since his third body (the first and second Doctors never managed a topless scene), so again, the evidence points towards the Doctor having two distinct sets of memories.