"Even if it's a happy day for me, I'm full of sadness," Umtiti said at a press conference, just a day after Thursday's truck attack on the French riviera city which killed at least 84 people.
"There were a lot of victims, I'm very sad for all the families who were affected by what happened yesterday."
The conference was preceded by a minute's silence at the Camp Nou in honour of the victims, with around 50 more people still fighting for their lives.
Barcelona on Tuesday concluded a deal worth 25 million euros ($27.7 million) to sign Umtiti from Lyon on a five-year deal.
The 22-year-old said it was a "childhood dream" to wear the Barcelona shirt.
"When I knew there was this possibility to come to this great club I had a lot of emotions, I openly shed some tears," he admitted. "It's really a dream to be at this club.
"I knew all the players, the style of play. I know almost everything about this club even if I wasn't around."
Umtiti made his international debut in France's quarter-final victory over Iceland at Euro 2016, retaining his place for the next two games as the hosts lost 1-0 to Portugal in the final.
However, he will face stiff competition for playing time at Barcelona from Gerard Pique, Javier Mascherano and France team-mate Jeremy Mathieu.
But Umtiti insisted the challenge will spur him on.
"I need this competition, I know it, and here there's a lot more competition," he said.
"I'm ready for that, I'm here to improve at everything and improve with these players who are more experienced than me. But I have lofty ambitions, I don't want to hide, I'm a player, I'm a competitor."