The stalemate puts Fleetwood six points clear of eighth-placed Plymouth Argyle with 12 games to play, although the Pilgrims have a game in hand.
County's Rodney Parade pitch played a part, with both sides missing chances from close range.
Newport had three early opportunities which all came from corners. Midfielder Lee Minshull glanced the ball wide at the back post from a Ryan Burge delivery before Tom Naylor and Rene Howe had similar efforts, neither of which forced Fleetwood goalkeeper Chris Maxwell into action.
But it was Fleetwood who had the best chance of the first half.
Jamille Matt squared the ball across the Newport penalty area and with only the keeper to beat, strike partner Antoni Sarcevic shot wide.
The best chance of the evening fell to Howe after 68 minutes but the Exiles striker misread the bobble of the ball after Chris Zebroski flashed it across the six-yard-box and saw his effort go the wrong side of Maxwell's post.
Newport now lie 10th, six points off seventh-placed Southend with two games in hand.