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.
Gary Warren was suspended for Caley Thistle. The central defender, who netted twice in the 3-2 win at Hearts on Saturday, was replaced by Josh Meekings - who had recovered from a back injury.
It was the only change for the hosts, while the visitors made two alterations to the side which drew 2-2 at home to Hibernian on Sunday. Attacker Michael Gardyne and defender Barry Douglas came in for Richie Ryan and Stuart Armstrong.
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.
At the opposite end Billy McKay forced the first meaningful save from Radoslaw Cierzniak. The Northern Irishman's header from the edge of the box came from another Doran cross.
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.
Daly headed over a Flood free-kick from six yards just before the hour, then Andrew Shinnie's low shot from a Doran cross was saved well by Cierzniak as the end-to-end battle continued.
Armstrong drilled a low angled shot wide, before John Rankin had his shot blocked and Armstrong fired the rebound over the target.
Cierzniak denied Andrew Shinnie once more when the Scotland international pounced on a Doran cross, but his header was saved comfortably.
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.