The 27-year-old has taken to the stand for the first time in his murder trial as the defence team's second witness of the day following pathologist Professor Jan Botha.
Pistorius denies murdering Steenkamp on Valentine's Day last year, maintaining that he mistook the South African model for an intruder when he fired the shots.
In court, he said in a trembling voice: "I want to apologise to Reeva's family, to those of you who knew her and are here today. There is not a moment and there hasn't been a moment since this tragedy happened that I haven't thought about your family.
"I wake up in the morning and you're the first people I think of, the first people I pray for. I can promise that when she went to bed that night she felt loved.
"I've tried to put my words on paper many times to write to you but no words would ever suffice."
Pistorius also revealed that he has been "scared" to sleep following the shooting.