The deadlock, played in 28 degree temperatures, kept each team winless at 0-1-2 records.
For the second straight match, the Fire opened the scoring and then were tied.
New York tied the game on McCarty's 13-yard shot off a scramble in the 21st minute. McCarty found the ball in a crowd after a corner kick was cleared by the Chicago defense, and beat Sean Johnson with a short right-footed kick.
Alex's hard left-footed shot for the Fire was saved by Robles in the 47th minute. New York flirted with a second score in the 64th, but Thierry Henry couldn't convert a pass from Dan Cahill in the penalty area.
Johnson made three saves, Robles two. The Red Bulls were cited for five yellow cards to Chicago's one.
The Fire was without defender Lovel Palmer, serving the first of a two-game suspension for a collision with Portland's Maximiliano Urruti in the previous match.