Justice Iqbal Hameed ur Rehman, one of three Supreme Court judges who were hearing the final appeal, said he could not take part.
"I was a part of the bench that was hearing the case of Salmaan Taseer, and this case is related to that," he told the court.
Salman Taseer, a Muslim, was a Pakistani politician murdered by his own body guard for speaking out against the country's strict blasphemy laws, which are often used to target Christians and other minorities.
Bibi, a mother of five, has been detained since 2010 and has not seen sunlight since last year.
She was convicted of blasphemy against Islam after an argument with Muslim woman over a bowl of water. She claims the allegations she offended the Prophet Muhammed are false and were made because of a personal grudge.
Lawyer, Khalil Tahir Singh, spoke to Premier after the postponement and said he did not feel safe and worried his car was being followed.
Security outside #Pakistan's Supreme Court before #AsiaBibi's hearing.6 years on death row for #Blasphemy charges pic.twitter.com/8Qjffvwx1A
— shaimaa khalil BBC (@Shaimaakhalil) October 13, 2016
Police in Islamabad were concerned violence would break out if Bibi had been freed. It was feared extremist Muslims would take their anger out on Christians and the judges involved in the case.
More than 100 police in riot gear were stationed at the court and hundreds others were on patrol in the city.
Andrew Boyd from Christian charity Release International, told Premier's Inspiration Breakfast: "Outside the court this morning there were 100 riot police - so you get a sense of the tension that is in the air and the real need for courage on the part of all involved in this."
It's not thought the final appeal will be heard in November. If she loses the hearing Asia Bibi will be hanged within 30 days.
Release International is calling on Pakistan to release Asia Bibi and repeal the blasphemy laws. But doing the right thing could well anger militants. Boyd said: "If that does happen then there are likely to be riots on the streets, Pakistan is well aware of that."
"And this kind of cases has been something of a poisoned chalice."
A British Pakistani Christian Association representative said that there are rumours going around that if Asia is acquitted and sent to the West, people will travel to kill her.
Mehwish, from the British Pakistani Christian Association, lives in Pakistan, and told Premier's News Hour a judge is not giving a verdict and has withdrawn from the case because he is scared of the consequences.
She said: "There's a risk on his own life, so he doesn't want to get involved in the Asia Bibi case, because whoever is going to announce any death penalty, or if he doesn't do that, they'll be a pressure [on them].
"So he has life threats [threats on his life]."
She also told Premier that there is now an even bigger price on Bibi's head, and that if she is freed she must leave the country immediately: "She can't stay here for a single second - I am sure about this.
"Some of the Muslim extremists have kept a reward for the person who is going to kill her... there's a reward for that special person who is going to kill her."
Mehwish said her fellow believers are "very, very disappointed" by the move.
"We were really expecting that this is going to be the last [case], but due to the delay everybody is so disappointed and I personally feel like this justice has crushed every limit.
"She has not done anything and every second she's praying that she is back to her family," she said.
Listen to Premier's Rosie and John speak to Andrew Boyd here:
Listen to Premier's Alex Williams speak to Mehwish here:
