Henry was withdrawn in the 42nd minute of Saturday's clash after going down following a challenge with Newcastle right back Kieran Trippier.
"Rico will have to undergo surgery in the next week and unfortunately he will have a long recovery period," said Brentford's chief medical officer Neil Greig.
The left back joined Brentford in 2016 and is the club's longest-serving player, with 203 appearances.
Brentford, who are 11th in the table, face Everton at home on Saturday.