In ‘Time Of The Doctor’ Clara accidentally makes the Doctor think she wants him to be her boyfriend. She quickly clarifies that she only wants him to pretend to be her boyfriend for the benefit of her family who are visiting. We should note that the Doctor doesn’t seem opposed to the idea, only mentioning he might have to check the manual as he is a bit rusty.
The Classic series never dwelled on the Doctor as a character who could get involved in romance. The 1st Doctor did get accidentally engaged in ‘The Aztecs’ but it is clear that he could never have got married. The 4th Doctor certainly had chemistry with Romana (no doubt thanks to the actors off-screen relationship) but it seems more a close friendship.
The 9th Doctor has an odd relationship with Rose, clearly trying to impress her and getting jealous of her relationships with others. He even mentions that he is capable of ‘dancing’. Despite this he doesn’t act on these feelings.
The 10th Doctor has feelings for Rose but thinks nothing of forming a deep emotional connection with Madame de Pompadour. He is also oblivious to Martha’s romantic feelings towards him. The only time he feels he can express his feelings is through a half human clone of himself.
The 11th Doctor is shocked at Amy’s advances (suggested to be because he still thinks of her as a small girl) and is annoyed at Clara’s suggestive comments, unsure if she is serious. The only time we see true passion and the suggestion of a fully developed romantic relationship is with River Song.
Could the Doctor ever have a companion in the future who he could have a romantic relationship with? Amy and Rory showed that a married couple could work in the TARDIS but would it be the same if it was the main character?
A major issue would be the Doctor’s awareness that he can never spend his entire life with a companion. From his perspective they would age and die very quickly, leaving him with two broken hearts.
This could be avoided if this hypothetical companion had a comparable lifespan or immortal. Practically, if this were to occur in the television show, this would require the casting of an actor who doesn’t age or the character would also need away to renew themselves (through regeneration or something similar).
Once this physical requirement is met is within his character to fall in love. The examples above show that he has the capacity, he just lacked the means. We’ve seen various examples of the kind of women that he likes. In particular he likes those who are passionate, intelligent, an ethical code he agrees with and a wonderment for the universe.
In theory his behaviour towards his companion/girlfriend/wife wouldn’t change and thus the structure of his typical adventures wouldn’t change. He would still be protective, still want to show them the universe and he would still have faith that they’d do what was right and help save the day.
The Doctor has also shown he is more than willing to follow the wishes of others. It would be an equal partnership, with his romantic interest driving the plot as much as the Doctor. Again, this doesn’t change the format significantly.
What could affect things is the usual turnover of new companions. Any new additions would have to be vetted by the Doctor’s partner. Usually the Doctor pays a lot of attention to a new companion but he’d have to make sure that he didn’t neglect his lover interest. This could make any new companion feel like a 3rd wheel.
Having a permanent partner might also change the Doctor’s priorities. While he has a new cycle of regenerations his life is still finite. He might not be so willing to sacrifice when it would mean leaving a loved one behind in mourning.
By the Doctor’s own admission he is a little out of practice at being a boyfriend. This could lead to rocky moments in his relationship and he could seek out past companions for advice.
This would allow adventures revisiting old friends to explore love and romance. How would they react to the Doctor (who might be in an incarnation they never met) talking to them about his new relationship?
If the Doctor was able to maintain a relationship what else might he have to revaluate? Could it be the impetuous to finally settle down and stop roaming the universe? Might he even think about starting a new family (if such a thing were possible)?
A lasting relationship would change the legend of the Doctor. His partner would become just as famous (or infamous) as the Doctor. This could make her very influential and put her at risk.
The biggest threat to the Doctor maintaining a relationship is regeneration. It has been established that he becomes a new man once he has transformed. There is no guarantee that a new incarnation would still be in love with his partner.
While tragic this does allow for a natural end of a plot arc focusing on the Doctor’s relationship. It allows for an epic romance to stretch through a single incarnation and then bring it to an end without making the Doctor or his partner seem like the villain.
Of course his partner might not take it well, becoming a villain. They could be a reverse River Song, targeting earlier incarnations. Could this be the origin of the Terrible Zodan?
This line of thought could be explored within your own games. It is an interesting personal dynamic that the show is yet to cover (aside from the hints about River Song) and could be interesting to see how it affects adventures.
With many new future incarnations of the Doctor to explore it is possible that this might happen at one point. PCs playing original characters might meet him or his partner during this period.
There is plenty of drama to be exploited with the PCs encountering this arrangement at different points in the relationship. What if they learn that the Doctor’s wife is going to die, betray him or worse? What if when they meet the Doctor it is still early days in their relationship and he is madly in love?