Following their defeat to Serbia and Montenegro in Glasgow in April, Britain must beat Israel to avoid going into a play-off match against Ukraine to decide who will be relegated from the Europe-Africa Zone into the Davis Cup's third division. The Israel match is being staged on grass at Devonshire Park in Eastbourne from 21-23 July.
Rusedski suffered the injury last month before the Stella Artois Championships at Queen's Club, where he withdrew in the first round while trailing to Antony Dupuis.
"I'd been hopeful of recovering from the injury," Rusedski said. "But after further examinations, I had to accept my consultant's advice to rest for the next four to six weeks. I wish GB the best of luck and look forward to returning to the circuit in good health shortly."
Jeremy Bates, the British Davis Cup captain, hopes to announce his team within the next week. Rusedski's absence will increase the pressure on Andy Murray, who will probably be asked to play in the doubles and two singles matches. With Tim Henman retired from Davis Cup tennis, Murray is the only British player available who is ranked in the world's top 100. To add to the problems, James Auckland, a doubles specialist, is unavailable for the tie because he is getting married that weekend.
Alex Bogdanovic, the British No 4, asked not to be selected against Serbia and Montenegro because he did not feel mentally ready. The 22-year-old lost in straight sets to Noam Okun when Britain won in Israel last year. His only previous appearance for Britain was in Sydney three years ago, when he lost in straight sets to Lleyton Hewitt before winning a dead rubber against Todd Woodbridge.
The British No 5 is Arvind Parmar, who won his first Davis Cup rubber at the seventh attempt when he won a dead rubber in Glasgow.