More than one million people are estimated to have lined the route of the stage, from Leeds to Harrogate.
Froome said riders needed "a bit of space" so they did not have to stop to file between lines of spectators.
His fellow Briton Geraint Thomas said: "It was great to race on home roads, but it is quite dangerous at times."
As defending Tour champion, it doesn't get much better than this
Froome, 29, finished the stage, won by Marcel Kittel, in sixth place while Team Sky colleague Thomas was 17th.
Welshman Thomas, 28, said: "It was so noisy. My ears are ringing now; it was like being in a disco for four hours.
"Everyone's going on about how good it was."
Froome, the second Briton to win the race after Sir Bradley Wiggins, said the crowds were "second to none".
He added: "As defending Tour champion, it doesn't get much better than this."
But compatriot Simon Yates warned fans must be careful if they are taking pictures with passing riders.
The Orica GreenEdge rider, 21, said: "The only real complaint is people taking selfies in the middle of the bunch and having their back to the peloton.
"Everyone is out there having fun and a lot of people don't realise we're coming past at 60kph (37mph) and we use every inch of the roads."
Mark Cavendish, the fourth British rider in the race, dislocated his right shoulder and suffered ligament damage after crashing near the end of the stage.
Sunday's second stage takes the cyclists from York to Sheffield.
Defending champion Chris Froome said the crowds for the Tour de France's first stage were "incredible" - but asked fans to give riders more space.