Robertson, 19, was recalled earlier this month by the Hoops following a loan spell at Rovers' League One counterparts Gillingham and has now linked up with Darren Moore's Doncaster.
The teenager featured 19 times for the Gills in 2020-21.
Robertson's arrival follows the captures of Omar Bogle, Josh Sims and John Bostock and is likely to be the last piece of inward activity in the winter transfer window.
Earlier this month, Rovers brought in loanees Ellery Balcombe and Elliot Simoes from Championship sides Brentford and Barnsley respectively, with the signing of Robertson taking their number of newcomers to six.
Robertson is eligible for Rovers' League One game at Lincoln City on Saturday afternoon.
Speaking after Robertson's recall by Celtic, Gillingham boss Steve Evans said: "Scott is an outstanding young player who has developed nicely in the last few months.
"He has returned to Celtic and we wish him all the very best.
"I cannot give any bigger testament to a in Scott Robertson than to say he will play in the Celtic first team next season."
Robertson made his senior Celtic debut aged 18 in a Europa League group stage fixture against CFR Cluj in December 2019 and has previously been a regular for the Parkhead outfit's under-21 side.
He is contracted at Celtic until the summer of 2023.