The 27-year-old had hoped to put his problems behind him after playing the full 80 minutes of Newcastle's 26-21 Guinness Premiership win over Bristol.
But Wilkinson has suffered pain since the game and medical examinations have shown damage to the kidney. He has been advised to rest for one month before resuming any form of training.
Robinson, currently preparing his team for Saturday's Test match against Argentina at Twickenham, said: "This is yet another unfortunate setback for Jonny.
"Everyone in the England camp wishes him all the best for his recovery.
"We are looking forward to seeing him back playing for Newcastle Falcons and putting this unlucky run of injuries behind him."
Wilkinson has suffered a succession of injuries since the 2003 World Cup final and has not represented England in three years. He was named in England's elite player squad earlier this year but remains rooted on 52 caps and 817 points.
Newcastle's director of rugby John Fletcher said: "We are all desperately disappointed for Jonny.
"He came back and showed fantastic form last Friday.
"It's a big blow for the team, especially with Toby Flood being away on international duty, but more importantly than that it's a big blow for Jonny himself.
"He has shown in the past what resilience he has in the face of injury, and I'm sure we will see the same amazing strength of character this time round."
Fletcher continued: "Jonny will have a month off from all training, build up from there and we are deliberately not giving an estimated date for his return to match action."
Wilkinson had made three starts for the Falcons this season, scoring 27 points.
Having started the season at Northampton he limped off in the home win over Worcester on September 8 with knee trouble, only making his comeback last Friday.
The news came the day after the Falcons discovered that number eight Phil Dowson will miss the rest of this season with a bicep injury suffered in last month's match at Wasps.
The 25-year-old has suffered a bicep tendon rupture, and will undergo an operation on Monday.