Yes, although he doesn't play by the usual rules. There are several examples of the Doctor being romantic with Rose ("Doomsday", for example), Astrid Peth (who accepted his invitation to be a companion before she died in "Voyage of the Damned"), River Song (who he married, of course, so several examples of this up to and including "The Husbands of River Song"), and Clara Oswald (several examples during the Eleventh Doctor era including "The Day of the Doctor", plus for the Twelfth Doctor "Mummy on the Orient Express" is an example). Eleven was also romantic with the Clara Oswin Oswald echo in "The Snowmen". The Eighth Doctor was also romantic with Grace Holloway in the 1996 TV movie, "Doctor Who". The Fourth Doctor and Romana were seen in romantic circumstances as well in "City of Death", however at that time such depictions were forbidden so it was more implied than anything else.
As mentioned above, the Doctor does not play by the usual rules of romance, in particular the Twelfth Doctor. In interviews with Wil Wheaton, Fox Germany and in convention panels, Peter Capaldi and Jenna Coleman have explicitly said the Doctor and Clara were in a romance, but of the "old-fashioned" variety, and neither were prone to tell each other how they felt directly. Coleman has said many times they express their feelings through what is "not said" such as looks and gestures. The episode "Under the Lake" and the Cloisters segment of "Hell Bent" are examples of this.