Some regions have been reimposing local measures to try to stop the spread of the fourth wave.
Cafes and restaurants in the southern city of Perpignan have to close at 11 p.m. for the rest of July and the obligation to wear face masks outside in public has been reintroduced in La Rochelle.
The prime minister said a regional approach was sensible for now and that no immediate national measures such as a "lockdown-lite" would be needed as long as the vaccination program continues to pick up pace.
He said that the government's aim was for 40 million first vaccinations to have been administered by the end of July and 50 million by the end of August.
As of Tuesday July 20, 38.2m people had received their first jab and 31.3 million people were fully vaccinated.
Castex warned that the government would be "intransigent" in the face of those who commit acts of violence.
His comment comes after an arson attack against one vaccination center last weekend and anti-vaccine graffiti was sprayed on another clinic.