The hosts pinned their opponents back for long periods but could not find a breakthrough.
County responded with an improved second-half display and but for goalkeeper Simon Royce, Jim Gannon's side might have gone through at the first attempt.
On a slippery pitch, Mark Stimson's side began at a high tempo and caused Stockport problems.
Their good start was almost rewarded but Mark Bentley headed wide.
The biggest talking point came on 25 minutes when Gills striker Simeon Jackson sprinted on to a through ball only to tumble under pressure from James Tunnicliffe.
As last man, the defender would have been sent off if the referee gave a foul but instead he waved play on to the disbelief of the home fans.
Fired up by the perceived injustice, Gills went on to play their best football of the season.
Stockport struggled to cope and had goalkeeper Owain Fon Williams to thank as he denied Adam Miller.
The interval came at a good time for Stockport, who switched from 4-4-2 to 4-5-1 to gain a foothold in the game.
It worked and they went on to enjoy the best of the chances before the final whistle.
The closest they came was in the 85th minute when Stephen Gleeson volleyed just wide from 15 yards.
A minute later the midfielder's low drive was saved by Royce at full stretch.
In the end a draw was probably the fairest result, but it left Gills wondering what might have been if they had taken the lead when in charge, or indeed if Tunnicliffe had been given his marching orders.