Messi had the chance to put Argentina ahead from the penalty spot, but his tame effort was easily kept out by Hannes Halldorsson as the spoils were shared in Moscow.
"We had a hard time creating chances," Messi is quoted by Mundo Albiceleste as saying. "We're going to go out there to beat Croatia [on June 21].
"We were superior [to Iceland]. We tried but it wasn't to be. Missing the penalty hurt.
"Worried? No, I'm angry and it hurts because I missed the penalty. I feel responsible for not having the three points. It would have changed everything.
"We are calm. We don't have to go crazy. We just started and we know it won't be easy. It hurts because we deserved to win."
Iceland failed to record a single shot in the second half, while Argentina had 16 attempts on Iceland's goal in that period.