The defender picked up a hamstring injury in Saturday's 1-0 defeat to Reading and hobbled off in the first half.
Scans have shown that it is a grade two tear, meaning he will be out until the international break at the end of the month.
It is a bitter blow to Robertson who has been beset by injuries during his time at the Millers and recently spent a couple of months on the sidelines with a broken foot.
It leaves boss Paul Warne scratching around for options at the back for Wednesday's crunch South Yorkshire derby with Sheffield Wednesday as Richard Wood is still a doubt and Michael Ihiekwe is suspended.
“He’s tweaked his hamstring,” Warne said. “When he did it I knew straight away we were in trouble.
“We took him off before he did any serious damage. Grade 1 it’s 4 to 6 weeks, so anything between two to four weeks I’d suggest.
“We just need to get to the international break and then we’ll have a lot more players back; Lindsay, Cheio, Icky will have done his suspension after the Wednesday game which will be a boost, but we just have to keep shuffling the pack, which makes the job more difficult.
“We’re just a bit on the side of bad luck.”
The Millers' plans for the match at Hillsborough were thrown into disarray on Monday after the shock appointment of Darren Moore at S6.
The Owls, who had been managerless since Tony Pulis left at the start of the year, raided Doncaster for the services of Moore.
Warne's side can go four points clear of Wednesday with a win.
He said: “I don’t think it’s a six-pointer, every time we play Sheffield Wednesday it’s a big game, with our positions we’re both desperate to get a win.
“It is a big game, it’s always been a big game, I don't think it gets much bigger than what it is.
“It’s a game where we have to pick our sharpest, freshest players and try and pick a positive team because we won’t go there to sit behind the ball, we’ll go there and try and get at them and win the football game.”