Modi, however, has refused to apologise as he maintains the comments are true.
He claimed last year he would produce evidence to back his claims, which he said was the reason the all-rounder was axed from the 2010 IPL auction list.
Cairns told the court the allegations were libel, AFP reported.
"The defendant's allegations have also had a profound effect on my personal and private life. It put a strain on my marriage. It hurts that my wife may think that I am not the man she thought I was," he said in evidence.
"It hurts me too that friends, many of whom are former cricketing foes, will question my integrity as a man and a sportsman and that all I achieved in the great game of cricket is dust."
Cairns represented New Zealand in 62 tests, 215 one day internationals and two Twenty20s for New Zealand. He made his test debut against Australia in 1989 and finished his international career with a Twenty20 match against West Indies in February 2006.
Modi's allegations refer to two seasons of the ICL when Cairns was captain of the Chandigarh Lions.
Six former Chandigarh teammates are to be called to testify against Cairns: Sharma, TP Singh, Love Ablish, Gaurav Gupta, Amit Uniyal and Karanveer Singh.
Former South Africa all-rounder Andrew Hall is also to be called as a witness for Modi, Cricinfo reported.
Cairns' counsel, Andrew Caldecott QC, said that in October 2008, Cairns was called to an ICL directors meeting in a hotel room where, according to the cricketer, his denial of rumours of his alleged involvement in match-fixing appeared to be accepted.
"No specific charges of match-fixing were ever put to Mr Cairns. No names of any accusers were mentioned.
"He was never informed after his departure that statements had been taken - mainly later - from Indian players alleging corrupt activity by various players including him."
Mr Caldecott said Cairns was then effectively suspended for not declaring an ankle injury which had been exacerbated by a 1000km charity walk in memory of his late sister.
He returned to New Zealand for an operation, was never contacted over any complaint and, in 2009, was entered for the IPL auction.
Modi's counsel Roger Thwaites QC, told the court Cairn's ankle injury was a "cover up" by Cairns and the ICL of the match fixing investigation, Cricinfo reported.
"You tried to leave with your tail between your legs in the hope that match-fixing would not leak out as the real reason for your dismissal," he said.
Mr Caldecott said the initial publication audience of Modi's tweet was up to 95 people while the initial publication of the Cricinfo article was between 450 and 1,500 people.
"Of course these receivers are bound to be cricket fans - it's the classic kind of allegation which runs and runs - one fan to another to another - match-fixing is a subject which is always topical.
"Be in no doubt that to everyone who hears it - Mr Cairns' entire career is tainted."
"Preserving the integrity of any sport depends as much on vindicating the falsely accused as it does on convicting the guilty."
The hearing is expected to last two weeks.
The case has been seen in the UK as the latest example of "libel tourism".
Padraig Reidy of the Index on Censorship told the Daily Telegraph the case was one of the "most clear-cut cases of libel tourism we have seen".
"While cricket is an international game, the alleged libel took place in India, concerned conduct in India, and primarily affects Cairns' reputation in India. The fact that he is pursuing the case in London proves that London's courts are seen as a soft touch, and that libel tourism is a very real phenomenon."
The UK Government is looking at tightening the rules to stop people from bringing libel action unless it could be shown their reputation had suffered "substantial harm" in England and Wales, the Telegraph reported.
- Herald Online, AFP
