The teams, who had shared 12 goals in their last two matches, cancelled each other out as the hosts squandered the chance to move clear in second place.
Motherwell leapfrogged the Highlanders with their 2-1 win at home to leaders Celtic, while United moved above Aberdeen to within one point of the top six.
Owain Tudur Jones was the first to threaten when his header from an Aaron Doran corner flashed past the post.
United responded with Jon Daly and Johnny Russell combining for the latter to shoot wide. Daly had a golden opportunity midway through the first half but he headed Willo Flood's cross wide from close range.
Andrew Shinnie might have made the breakthrough in the final minute of the half but his volley from the edge of the area was turned wide by Cierzniak.
United substitute Armstrong tested Inverness goalkeeper Ryan Esson with a shot from distance at the start of the second half - but the stopper was equal to it.
Armstrong drilled a low angled shot wide, before John Rankin had his shot blocked and Armstrong fired the rebound over the target.
Douglas sent a 25-yard free-kick wide from a central position midway through the second half.
Esson excelled 12 minutes from time with a smart save. Rankin's shot from the edge of the area was deflected off Daly en route to goal - but the stopper reacted smartly to grasp the ball on the line.
Caley Thistle club captain Richie Foran replaced McKay to make his return from a knee injury, which had him sidelined for the last four weeks, with nine minutes to go.
The Irishman forced a late save from Cierzniak when he connected with a David Raven cross. Caley Thistle weathered a stoppage time onslaught to secure a share of the spoils.