The 55-year-old tweeted from the hospital in Tooting to thank the professionals who "saved his life".
Moore won 64 England caps between 1987 and 1995, played for the British and Irish Lions, and trained as a solicitor while pursuing his rugby career.
The former England hooker now works in the media.

The suspect was tracked down at the Underground station for Heathrow Terminals one, two and three.
Earlier, armed officers carried out a search on a Victoria line train at King's Cross St Pancras Underground station following reports of an incident at Finsbury Park.
The victim suffered cuts to his body and face and was taken to hospital.
British Transport Police said his injuries were not thought to be life-threatening.
A spokesman told the BBC armed officers attended because they had been on patrol nearby.
The Victoria line was suspended in both directions for around 15 minutes.  It has since reopened but is severely delayed, a Transport for London spokesman said.
Liam O'Hare, who was at Finsbury Park at the time of the incident, told BBC London he saw an altercation between two men on the southbound Victoria line platform.
He said: "One of the guys ran off in a different direction, and one boarded the train but was immediately helped off as he was clearly injured.
"He was bleeding, there was a puddle of blood on the train floor and people had to step over it."
He added that other passengers reported seeing one of the men holding a blade, but said he had not seen it himself.
Former England rugby international Brian Moore is recovering in intensive care at St George's hospital in London after suffering a heart attack.
[NEXT_CONCEPT]
A man has been arrested on suspicion of grievous bodily harm after an assault on a 63-year-old man at Finsbury Park.