The 23-year-old former Arsenal player was taken to hospital after falling to the turf at White Hart Lane just before half-time. He remained in a critical condition hours later, the hospital said in a joint statement with Bolton.
No other player was involved in the incident and after Muamba had been treated for several minutes by medical staff, referee Howard Webb took the players off the field.
The score was 1-1 when the game was abandoned.
The 6ft 2in Muamba, who was born in the Democratic Republic of Congo but has represented England at every youth level from under-16 to under-21, was treated by medics on the pitch.
A Reuters eyewitness said he was given CPR (cardiopulmonary resuscitation).
A number of players gathered around Muamba, some crying and holding their heads in their hands.
As he was carried from the field on a stretcher there were a few bursts of applause from the crowd before they fell silent.
In the earlier quarter-final, Everton and Sunderland drew 1-1 at Goodison Park.
Sunderland, without a win over their opponents in any competition since 2001, started well and it came as no surprise when Scotland defender Phil Bardsley gave them a 12th-minute lead with a rare goal.
A free-kick wide on the right was touched to Bardsley who took a couple of steps before unleashing a powerful long-range drive that eluded a cluster of players in the penalty area.
It was the full-back's fourth goal since joining twice FA Cup winners Sunderland from Manchester United four years ago.
Everton, who have lifted the trophy five times, equalised against the run of play in the 24th minute.
Tim Cahill, who has notched more than 100 goals from midfield in his career, has struggled to find the net in recent months but his second goal of the season set up a replay at the Stadium of Light on March 27.
Cahill beat Belgian international Simon Mignolet from eight metres after showing quick reactions to nod in a header from Nikica Jelavic.
The Australian had two chances to grab the winner for Everton before Mignolet flew to his right in the closing stages to save a flicked header from John Heitinga and quickly picked himself off the floor to stop Jelavic's close-range followup.
"That was a fantastic double save by Mignolet, really brilliant," Sunderland manager Martin O'Neill told ITV Sport.
"I thought we started off really well. Their equaliser knocked us back a bit and after that the result could have gone either way."
Everton captain Phil Neville was frustrated his side did not seal a place in next month's semi-finals at the first attempt.
"We wanted to make use of home advantage by winning today and we put everything into that game," he said.
"We didn't start well but we came into it at the end and we're disappointed we didn't capitalise on our chances."
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