Did The Doctor ever meet Doctor Who from the TV series?

In 2012 comic, The Girl Who Loved Doctor Who, The Doctor finds himself in our universe, where the show exists. That's the most similar example I can think of.

Here is an answer based on the TV series. The 7th Doctor serial Remembrance of the Daleks is set in 1963, the year in which the first episode of Doctor Who was broadcast. It contains a scene in which an announcer on a television says:

This is BBC television, the time is quarter past five and Saturday viewing continues with an adventure in the new science fiction series Doc—

but someone switches the television off before the sentence is finished.

This isn't definite proof of course - it could be that Doctor Who takes place in an alternate universe where there was a different new BBC science fiction series in 1963 whose name began with "Doc" - but it does suggest that the show Doctor Who exists in the Whoniverse, in the 1960s at least.

On the other hand, though, one of the early episodes of the Christopher Eccleston era has a character search the internet for "doctor blue box." He comes up with lots of conspiracy stories about a mysterious time traveler, but nothing about the TV show Doctor Who. This suggests that in the New Who continuity the TV show doesn't exist. I guess we'll have to put it down to disruption of the space-time continuum due to the time war or some such. (Long-term continuity has never been a strong feature of Doctor Who.)

Sort of, but this is weird

Virgin Publishing's The New Doctor Who Adventures Book 43 'Head Games' featured a fictionalized version of the Seventh Doctor in our universe, known as 'Dr. Who'.

This plot summary explains the convoluted mechanism for this happening:

A flaw in the structure of the Universe is allowing energy from the Land of Fiction to seep through. Things get even more complicated when the fictional energy finds a focus in Jason, the young Writer who was returned to Earth by the Time Lords after the Doctor's last encounter with the Land of Fiction. As the fictional energy floods into this Universe, Jason finds that his wishes and dreams are coming true. A fictional double of the Doctor, Dr. Who, appears in the TARDIS, knocks out the real Doctor before he can enter the crystal, and sends him to the fictional Galactic Prison for the crime of trying to wipe out the Detrians. Dr. Who then picks up Jason, his new companion, and they set off to have neat adventures, beat up green monsters, and arrest the evil Doctor's accomplices.

Critic Finn Clark wrote:

The Land of Fiction stuff is present, but more in the background than I'd expected. Instead the book's more interested in 'Dr Who' [...] Overall I enjoyed Head Games. It contains massive misjudgements, but they're more palatable these days. Worth a second look.