Jayasuriya, 36, needed several stitches on his right thumb that was split while catching Pakistani opener Imran Farhat in the gully region on the second morning of the Test on Tuesday.
"It looks like he will be out for at least four weeks," Sri Lankan team's physiotherapist Tommy Simsek told reporters after returning from the Kandy Central Hospital.
"There's also a dislocation near the joint. It is very unlikely he will take any further part in the Test."
The former Sri Lankan captain announced last week he will retire from Test cricket after the second Test, but will continue to play one-day internationals.
The Kandy Test was his 102nd match in which he made 14 in the first innings.
He scored 6,613 runs with 14 centuries and a best of 340 against India in Colombo in 1997. He also claimed 92 Test wickets.
Jayasuriya, who is Sri Lanka's highest run-getter in both forms of the game, has scored 10,625 runs in 357 one-day internationals with 19 hundreds and 273 wickets.