n a statement, the reality TV star said: “I am truly very sorry for the grief I have caused them.”
He added that “the vast majority” of the allegations reported were “untrue and unfounded”.
The UK premiere of The Blind Side, due to be held next week, was scrapped after Oscar-winner Bullock pulled out citing “unforeseen personal reasons”.
“There is only one person to blame for this whole situation, and that is me,” James, 40, said.
“This has caused my wife and kids pain and embarrassment beyond comprehension and I am extremely saddened to have brought this on them. I hope one day they can find it in their hearts to forgive me.”
He added: “Beyond that, I will not dignify these private matters with any further public comment.”
Allegations of an affair surfaced earlier in this week after In Touch magazine published an interview with tattoo model Michelle McGee who claimed to have had an 11-month relationship with James while Bullock was filming The Blind Side – a role which earned her best actress at the Oscars.
A spokesman for the actress would not comment on Miss McGee’s claims.
Bullock, 45, and James met in 2003 before marrying in 2005.