The result marked the Gulls' third successive goalless draw at Plainmoor, and moved them up to sixth place ahead of the rest of the weekend's action. Promoted leaders Chesterfield are now three points ahead of title challengers Bury, who host third-placed Wycombe on Saturday.
Guy Branston came close to scoring in the fourth minute when his powerful header was headed up off the goal-line and on to the bar following Kevin Nicholson's corner from the left. Branston again hit the bar from another Nicholson corner midway through the second half as United tried in vain to get the breakthrough.
After bossing the early stages, the Gulls had to focus when Chesterfield top scorer Craig Davies beat marker Chris Robertson wide on the left and then let fly with a 30 yard dipper. Minutes later, Davies then broke down the right channel and sent a dipping shot on the run towards goal that home keeper Scott Bevan was forced to tip over.
United countered with Jake Robinson firing a cross across the face of goal and then defender Simon Ford blocked a Billy Kee strike on the 18 yard line.
Bevan made a superb save to deny Derek Niven from 20 yards, and then made another excellent block stop to deny forward Deane Smalley from close range.
United started the second half as they finished the first with Kee's volley screaming just over the bar, before Branston rattled the woodwork once more. In stoppage time, Chris Zebroski flashed a header just over from Craig Stanley's corner from the right.