Wasn't Iron Man smart enough not to fight the Winter Soldier?

Isn't Tony Stark smart enough, though, to know that this is exactly what the bad guy wanted him to do? Wasn't that obvious enough?

Of course. This is obvious to everyone, including Iron Man. But that's not the important part. We'll get into this more after the next bit.

Also couldn't the Winter Soldier even try to convince him to settle that later since they were still on a mission?

Bucky could try to convince Iron Man to settle it later, but if you think Iron Man is unstable right now, try being Bucky. He not only got to watch himself kill innocent people, but he has to remember doing it too, against his own will. There's undoubtedly a part of Bucky that thinks he deserves the revenge that Iron Man wants. The part of him that desires survival just happens to be a bit stronger.

But now to the deeper bit. Consider the following points:

1. What mission? They came to kill the other Winter Soldiers, but that's already taken care of. Mission accomplished. He has no need of Bucky's help anymore.

2. Why not kill Bucky? Iron Man already has plenty of reason to get rid of him. He's a wildcard. He's not his friend, not an Avenger, Iron Man owes him nothing. Bucky still has the potential to lose control of himself and kill more innocent people. Iron Man has 0 reasons to care about keeping Bucky alive.

3. Tony Stark, remember he's a human, just watched his parent's gruesome murders. He's already very emotionally unstable throughout the entire movie, why is it hard to believe that this would push him over the edge of all reason?

4. He says it right there in the move: "I don't care." He knows he's being stupid, but he just doesn't care. Sometimes people know that what they are doing is wrong, and they choose, consciously, to do it anyway.

I know it can be hard to see our heroes do stupid things, but that's what makes them heroes in the first place: The fact that they aren't perfect, that they make mistakes, that they do stupid things, and yet still rise up to overcome their humanity and do amazing things.

To me, this movie was all about reminding us who these people are. They aren't perfect, they aren't always right, nor always smart. And that's OK.

Revenge makes people blind.

Same happened to Iron Man, he became blind in revenge. From his first appearance in the film, it's shown how much he is affected by the loss of his parents and how much he feels guilt of what happened during their last conversation. So it was expected from him and Zemo did analyse him before making this complex plan.

If you even analyse his character then he had done stupid things before, that's also what made him more human, as humans do make mistakes. Emotion controls us more than our brain. It was clear from his conversation too.

Steve Rogers: It wasn't him, Tony! Hydra was in control of his mind!
Tony Stark: MOVE!
Tony Stark: Do you even remember them?
Bucky Barnes: I remember all of them.
Steve Rogers: This isn't gonna change what happened.
Tony Stark: I don't care. He killed my mom.

So he was just blind in revenge for the death of his mom.

Similarly Black Panther also became blind in revenge but he did realize his mistake later on.

Tony Stark is certainly a "smart" man, as evidenced by the amazing technology that he's created. But he is still just that: a man. No matter how much of a superhero he is, or how powerful he is, or how famous he is, Tony Stark still has emotions.

Consider what happened. His parents were killed. Murdered. By the bloke standing right in front of him. I don't think it's difficult to believe that all reason went out of the window, to be replaced by pure, unbridled rage. These were his parents!

One might also argue that, by virtue of the very nature of his job and of all the responsibilities weighing down upon him — not to mention the extraordinarily stressful and traumatic events of both Iron Man 3 and Avengers: Age of Ultron, Tony Stark is far more liable for an emotional breakdown than you or I.

In short, being "smart" has nothing to do with it.

_{To be quite honest, I didn't fully buy into these scenes. I found it difficult to believe that Stark would act so illogically but without too much in the way of emotional outbursts. That could be an issue with the movie's direction because, if anything, I would have expected Tony to be a lot more visibly out of control than he actually was.}

We're mostly thinking of Tony Stark as the superhero Iron Man. In this case, Black Widow's review about Iron Man/Tony Stark in Iron Man 2 movie is correct.

Iron Man is Tony Stark, the one who thinks of what the best course of action would save the most people and is not afraid to sacrifice his life for the safety of other people.

On the other hand, Tony Stark is a genius, billionaire playboy orphan with father issues and betrayed by his father figure for his arc reactor (and recently, by his childhood hero Captain America). Though his heart is in the right place, he's the emotional equivalent of a seven year old kid. He's also very human and finding out that his parents were murdered, it's natural to want to avenge their death (so Batman there) and/or bring justice and give Tony and his parents closure. I don't think there is a person out there who could stay calm when confronted by their loved one's killer - no matter if the murderer doesn't know about it.

With due respect of all above opinions, it is highly possible for a person whether a superhero or not, to lose all sense of appropriate thinking, because he is a son.

In case of Tony Stark, He never adored the moments he had with his parents and in his twenties, he lost his parents long before his captivity by the Ten Rings in Iron Man 1, when he learned to drive his responsibilities correctly, also he missed his parents as well.

Now in Captain America: Civil War, when he came to know of his parents' death to be a brutal murder not merely just a car accident and the assassin is right in front of him (not his friend like Captain America) what should one expect him to do? After all he is still an Avenger, avenging his parents' death.

And about Bucky Barnes, in the post-credits scene Bucky chose to go into cryo-sleep because even he knows that he could be a danger to the society, a lethal weapon.

I think Tony Stark did what one has to do after knowing the truth. Really in person, after learning about his parents' death, everyone can give the same reaction. And I also appreciate Bucky Barnes for returning to Cryo-sleep, which is safe for him and for others as well.