Fathers4Justice in David Cameron doorstep protest
He wants Mr Cameron to honour pledges about grandparents having a right to see their grandchildren and over shared parenting.
Mr O'Connor, from Hampshire, said he would stay as long as possible.
He also wants Mr Cameron to retract comments he made describing some fathers as "runaways".
Mr O'Connor said: "The idea is to carry on for as long as I can - even if I am hospitalised."
Mr O'Connor said his only intake would be water and lemon juice.