Obama didn't want the mother to be burdened with having to abandon a baby that she never wanted. of course fathers have no say in whether they are obligated to that baby for the next 18 years after pregnancy occurs. It's a given that they are and the State comes down on the citizenry to prohibit all men from obtaining driver's licenses, etc. without proving they are not deadbeat dads.
I believe in the video he was talking about the necessity to bring in another doctor other than the abortionist to assess the baby. You are right that he also directed the word “burden” to the mother, parents or society who may now be responsible for additional unanticipated medical expenses.
From what we heard last night - he clearly believes it is the duty of the baby and the medical staff to allow the baby to expire without any attempt to secure life and the blessings of liberty for him/her.