Broad took to Twitter to apologise for the heavy defeat – and to take a swipe at ex-players for being too critical in their analysis of the tourists' display.
The 26-year-old, who failed to take a wicket in his 24 overs in India's first innings and was not given a bowl in the brief second, posted: "India outplayed us. A few positives but on the whole poor so sorry for that. Onwards and upwards.
"And before u listen to too many ex playing 'experts' being negative, ask them if they ever won a Test series in India....#28years."
England have not emerged triumphant from a Test series in India since 1984-'85.
Botham – part of Sky's commentary team for the match – took umbrage over Broad's attack on former players and hit back.
In a tweet to Broad, he wrote: "Didn't average 40+ with the ball overseas...! Not sure what I scored against India with the bat..? #justsaying."
Botham's record with the bat in India was impressive. He hit 114 in the Golden Jubilee Test in Mumbai in 1980 and averaged 55 as England lost the six-Test series 1-0 in 1981-'82.
When another Twitter user tweeted both men to say, "Ian should not kick a man when he is down me thinks!!", Botham retweeted, adding: "Who started the kicking..?"
Botham had earlier suggested on television that Steven Finn coming in for Broad should be one of four changes for the second Test, which gets under way on Friday.
Broad later took to the social networking site to play down a potential brewing row and hailed Botham's playing achievements.
"Spat with @Beefy Botham?? What spat? Love Beefy, best bowler England have ever had, then add his batting. Twitter is funny at times!" he tweeted.
Given his indifferent form, Broad's spot is likely to come under scrutiny after India powered to victory yesterday.
England captain Alastair Cook took comfort from the character his side showed in the second innings with the chances of a brave rearguard being commuted into a famous stalemate in this first match of four centred on Cook (176) and Matt Prior (91) on the final day.
But they could augment their combined defiance by only 16 more runs yesterday morning, and with their stand of 157 broken, England lost their last five wickets for only 50 in a lunchtime 406 all out.
India's resulting target of 77 was then treated with near contempt by Virender Sehwag and Cheteshwar Pujara, whose aggression put paid to any fanciful notion that Graeme Swann's off-spin might yet make life difficult on a worn pitch.
Cook knew he and Prior had to remain at the crease if England were to pull off something special but he was pleased with his team's fight.
He said at the post-match presentation: "There was an outside chance: Matty and I knew we would have to do the majority of the work. We got closer and closer and knew India had done a huge amount of overs in the field as well."