Finn has suffered a strain to a disc in his lower back and will be analysed before the start of Thursday's Test in Nagpur.
Twenty20 captain Broad may yet play in the T20 internationals in Pune and Mumbai next Thursday and Saturday as he remains in India to undergo treatment.
He was highly unlikely in any case to be selected to play India in Nagpur, when the tourists will be bidding for a historic 3-1 series win.
The fast bowler lost form, after first suffering a heel injury in a tour match in Mumbai last month, and was not involved in last week's seven-wicket victory at Eden Gardens as Finn returned to fitness and replaced him.
After suffering more discomfort at net practice on Wednesday, Broad was sent for a second scan - which showed exactly the same as the first one he underwent five weeks ago.
Eoin Morgan is England's Twenty20 vice-captain and would be the obvious candidate to lead the team in Broad's absence.
The selectors may, however, need to replace Broad in an inexperienced 13-man squad which contains only three other specialist seamers.
Test captain Alastair Cook confirmed Broad's absence at his press conference at the VCA Stadium.
"Stuart is out of this Test match," he said.
"His scan has showed a bruised heel, and we'll assess him over the next day or so to see what happens with the Twenty20. We'll know a bit more when it settles down."
Broad has endured an especially miserable tour, in which he did not manage to take a wicket in either of the first two Tests - and was then dropped for the third, in order to accommodate fit-again seamer Steven Finn.
"It's the nature of sport, isn't it? Some people have good tours, and some people don't," added Cook.
"Unfortunately, more down to niggles and and illnesses, Broady hasn't quite managed to get into the tour.
"It's frustrating for him, and disappointing for us as a side. But we all know the class of Broady, and he'll be back."