To be honest, I think Iron Man is getting up there among the most popular. Though its mainly due to the Iron Man movies, RDJ did a great job in portraying a character that once was more of an after thought in the past, compared to Spider-Man, Thor, and even Capt. America.
As for being up there with Spidey, Batman, and Superman? I'm not sure about that yet. Spidey is still the face of Marvel along with Wolverine. Spiderman's story remains one of the most well-known in pop culture: a kid who was given a gift/curse and has to endure the hardships of being a superhero.
Superman and Batman are icons for decades and have continued to be so even during these changing times. I'm even more impressed with Superman's perseverance as an icon as comics moved towards a more darker and more serious tone, which brought Batman to even greater popularity that surpasses Superman during the 80s till now. Sure, Supes is probably not the most popular hero due to the misunderstandings of him being too much of a boyscout, but his logo and name still resonates for a reason.
Anyway, I think Iron Man is just behind Capt. America, Wonder Woman (though some would say she's up there with Supes, Bats, and Spidey due to her own iconic status and enduring character), Wolverine, Robin, Hulk, etc.