John was a pilot in the United States Marine Corps. He was on active duty while House was growing up and, as a result, the family moved around a lot. It is implied that, as a result, Gregory was very close to his mother, but had few other friends and limited contact with his father.
John had what his son described as an "insane moral compass" - he always said what was on his mind and never lied to anyone. He is the antithesis of Gregory's credo "everybody lies". It is implied that this meant that John rarely bent the truth in order to comfort his son, and usually found fault with what Gregory did despite his usually excellent academic accomplishments. It is clear Gregory feared his father's punishments, that those punishments were doled out for even minor offenses, and that they were far better suited to Marine recruits rather than a talented young boy. In the Episode One Day, One Room, House admits that his father abused him by making him sleep in the yard or take a bath in ice whenever he misbehaved. Gregory also noted that his father got along with peers and superiors, but made no effort to get along with subordinates, including his son. House also felt that John cared more about his job than his relationships.
The relationship was further strained when the prematurely bright Gregory suspected at the age of 12 that John was not his biological father. John was away from home at the time Gregory was supposed to have been conceived, his father's second toe is longer than his big toe, while Gregory's is not (a genetic trait), and Gregory shares with a family friend a birthmark in the same location. Gregory confronted his father about it, but his father responded by not speaking to him for an entire summer. Gregory confirmed his suspicions after his father's death by obtaining a DNA sample and performing a paternity test. This suspicion has affected his views on relationships ever since.
As a result, Gregory was rather estranged from his father. His behavior around his father was clearly different from his behavior around everyone else. Around his father, Gregory was quiet, reserved and polite. Part of this may be more of an effort to avoid upsetting his mother. Gregory usually does everything he can to avoid being with his father, including skipping family events, and making up excuses to be elsewhere when his father is in the vicinity. House unwillingly attends John's funeral in Birthmarks and gives a short unflattering eulogy.
In the episode Brave Heart, House, who had recently moved in with Wilson overhears Wilson talking to his dead girlfriend Amber Volakis, this inspired House to try talking to his dad. That night, House said that he might have been focusing on the wrong parts of their relationship and maybe he wasn't such a bad father. House told Wilson that he thought what he was doing was stupid. Wilson told Amber that House has changed.
In the episode Perils of Paranoia, it is revealed that Gregory keeps his father's M1911 handgun and ceremonial Mameluke sword on the top shelf of his closet.
TV Tropes includes John House as an example of the Abusive parents trope.