The National Transportation Board doesn’t want you calling or texting behind the wheel and has called on states to ban the use of cellphones and other gadgets while driving.

The move is sure to bring howls of protest from just about every quarter, as a recent study by the National Highway Traffic Safety Administration found that, at any given time during daylight hours, as many as 13.5 million people are fiddling with a phone when they should be driving.

The recommendation, unanimously approved Tuesday by the five-member board, would apply to hand-held and hands-free devices used by the drivers...