Naomie Tremblay-Trudeau was taking part in an anti-austerity demonstration outside the legislature in Quebec City last week when she was shot at point-blank range.
The investigator has 180 days to file a report.
A spokesman for the Quebec City police has said Tremblay-Trudeau was actually struck by a cardboard cap used to retain the gas and not the canister itself.
Francois Moisan said police fired off the gas to keep protesters at bay.
Demonstrations aimed at getting the Couillard government to roll back austerity measures have become nearly daily events throughout the province.