The Polish-registered van struck one of three trucks blocked by the log barrier at around 3:45 am (0145 GMT).
The driver has not yet been identified "given the state of the body," a police source said.
It was the first death of a driver since the migrant crisis at France's Channel ports began in 2014, although there have been numerous fatalities among migrants seeking to hitch a ride to Britain by train and truck.
They could face charges of manslaughter, impeding traffic and endangering the lives of others.
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Interior Minister Gerard Collomb said he would head to Calais shortly, adding: "We cannot allow this illegality to continue."
Those responsible "must be taken to court (because) whether you are a migrant or French, you must obey French law," he added.
Two trucker associations called on local and national authorities to "take the most appropriate steps to deal with these problems and avoid a new tragedy."
David Sagnard of the National Road Transport Federation said in a statement: "We warned that this could happen, and it has happened... It's now urgent for the police presence to be reinforced."
Before Calais' infamous "Jungle" camp was closed last October, migrants desperate to reach Britain as stowaways on trucks frequently set up makeshift roadblocks, usually at night, to slow cross-Channel traffic.
The squalid settlement was home to up to 10,000 at its height. Between 400 and 600 migrants are today thought to be living in the Calais area in precarious conditions in the hope of reaching Britain.
Roadblocks began reappearing in late May with a new uptick of migrants in the region -- and a surge for Europe, with Italy registering more than 65,000 arrivals since January, up nearly a fifth from the same period last year.
More than 1,800 people have died attempting the crossing since the start of the year, according to IOM figures.
In early June Collomb vowed to prevent migrant settlements from springing up, announcing the deployment of three additional mobile units comprising more than 150 police officers.
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