A BOE (official state bulletin) published today states that travel across borders (by whatever means, land sea or air) must have one of the following:
Certificate showing receipt of both vaccines at least 2 weeks before.
Certificate of acceptable Covid test
Certificate of recovery from Covid
For coming into Spain, a QR code must be obtained by completing an FCS (health control form) and all those over 6 years of age must have one.
This is for citizens of all countries including EU countries (except for a few exceptions) and will take effect from Monday 7th June.
Of course, each country can impose its own restrictions so you'd need to check with the French gov sites to see if anything is different.
Edited. Those certificates will not be required when entering UK as it is one of the countries that is not on the at risk ones. France, however, is on the at risk list.
This message was last edited by mariedav on 05/06/2021.