No, there is no rule - the nature of the game makes tucking the ball away less practical.
Other than plays where the QB pitches to a running back, it is rare for a ball carrier to attempt to pass the ball to a teammate. In rugby, the ball carrier is always aware of teammates around him and is looking for an opportunity to unload the ball. This can be done to either side, so it is important that keep the ball in available.
When an American football ball carrier is tackled, he needs to hold on tight to the ball to avoid a fumble. In rugby, the ball carrier MUST release the ball when tackled, if he doesn't his team gives up a penalty. So again, if the ball is tucked in tight at the elbow, it can be much more difficult to release the ball.