Starfleet would surely need a lingua franca for its members to function dependably, and English is most likely that. It's probably required coursework at the Academy.
Among Federation citizens, I doubt the existence of the universal translator significantly affects their cultural identity. In societies largely free of scarcity, learning is one of the only things left to do.
Does Picard actually speak French all the time?
In the episode "Code of Honor," Data makes a comment that could lead the viewer to believe the French language is as dead as Latin. Picard's flawless English, English accent and affinity for English literature point to that as well. French is a dying language today.