4 July 1928: birth of Paddy Russell, who directed The Massacre (First Doctor, 1966), Invasion of the Dinosaurs (Third Doctor, 1974), Pyramids of Mars (Fourth Doctor, 1975) and Horror of Fang Rock (Fourth Doctor, 1977) - a pretty good record.
4 July 1973: birth of Marcus Wilson, producer and series producer for Series Six and Seven of New Who featuring the Eleventh Doctor.
ii) broadcast anniversaries
4 July 1964: non-broadcast of 'Hidden Danger', the fourth episode of the story we now know as The Sensorites, because of the gripping events at the third test match of the 1964 Ashes series (idf you're interested, Australia took five wickets in the course of the day, and another five when the second innings resumed on Sunday; England were unable to make up the difference and so lost the Ashes as per usual), also the Wimbledon women's final between Maria Bueno and Margaret Smith (later Margaret Court) which Bueno won 6–4 7–9 6–3 (Smith having defeated 19-year-old Billie-Jean Moffitt, later Billie-Jean King, in the semi-final - revenge for their match two years earlier having gone the other way when Smith was the world's #1 seed). Aren't you glad I told you that?
iii) dates specified in-universe
4 July 1927: in a flashback in the 2011 Torchwood episode Immortal Sins, Jack Harkness goes back to New York and has sex with Angelo Colasanto.