The 32-year-old from Victoria was taken to hospital after crashing 150 kilometres into the 205-kilometre stage from Montreux to Meringen.
"Hesjedal suffered multiple, severe contusions and abrasions to the right wrist, hip and knee and left shoulder, elbow and knee," his team, Garmin Sharp, said in a statement.
"Preliminary CTs show no fractures and no neurological pathologies. He will remain under close observation by team medical staff. Hesjedal's health is the team's priority — depending on how his recovery progresses, team medical staff as of now hopes he may be able to train in a few days time, although this will depend on his recovery."
The Tour de France begins June 29.
Hesjedal had been doing well in his comeback race after failing to defend his Giro d'Italia title last month when he was forced to withdraw due to an upper respiratory infection. The Canadian was in second place overall after Stage 2.
Peter Sagan of Slovakia led a four-man breakaway to win Monday's stage.
Sagan edged defending champion Rui Costa of Portugal and Roman Kreuziger of the Czech Republic to complete the stage in four hours 46 minutes 27 seconds. Local rider Mathias Frank had the same time in fourth.
Frank, riding for the Swiss-backed American team BMC Racing, climbed from third overall to lead Kreuziger by 23 seconds. Costa is third, trailing Frank by 35 seconds.
The race lasts nine days.
The fourth stage Tuesday is a rolling 174.4-kilometre route from Innertkirchen to Buochs.
It's been a tough year for last year's big winners on the world cycling stage.
Britain's Bradley Wiggins, who won the 2012 Tour de France, has been ruled out of defending his title due to a knee injury.
Wiggins was also forced to pull out of the Giro with a chest infection.
Garmin Sharp announced Friday that Hesjedal has re-signed with the team through 2015.