It applies to most (fictional) relationships that mortal peril situations increase the chance of getting "emotional" (and by that I mean romance-fuelled moments) by 1000%. Or some other high percent quantity. Of course, it's only logical. The moment one of the individuals is engaged in something life-threatening, the other one reacts by showing feelings and reactions he wouldn't normally indulge him/herself with in public. This also applies to Roy and Hawkeye.

And in their particular case, the "mortal peril situation" sindrome applies both ways and it's just as strong for both of them. Despite Hawkeye being the one who's supposed to protect her superior, Roy would go to any extent to protect any of his subordinates, and if it's her we're talking about...

Hawkeye: Why did you show up??!! You should've stayed behind no matter what happened to us! The enemies will associate you with us now! Bringing yourself here was quite stupid!
Roy: Aaah! I know, I know, I'm an idiot!! [...] First lieutenant.
Hawkeye: Yes, sir.
Roy: I'm glad you're alive.
Hawkeye: I'm sorry I worried you.

Hawkeye got attacked by Gluttony while talking with Roy on the phone. In a flash, he got a car, rode to where they were and saved both Hawkeye and Fuery from the homunculus. Of course, she was not too happy about it... Roy had just run out of the office, unprotected, and exposed himself to danger without a second thought.

Even if she told him everything was okay on the phone, communication being cut isn't the best of signs for Roy. It was the phone what told him Hughes was dead, after all. However strict may Hawkeye be about Roy's protection, she must have known he'll fly their way the moment he thought anything was wrong (that's probably why she tried to calm him down at first). And so he did, and got a scolding for it. Of course, the moment they're out of reach from Fuery's ears, Roy lets her know that he's relieved and she excuses herself for having worried him. Aah, the intricate art of faux miscommunication.

     

A little later, all of this would look like a child's play. Roy and Hawkeye split up while chasing after Barry the Chopper and his real body. At some point, Hawkeye and Alphonse encounter Lust, who tells her that she "will be joining her superior soon". Hawkeye believes her words (that Roy is, in fact, dead), and starts shooting mad with rage at her. Once she's out of bullets (and Lust is still unharmed), she starts crying hopelessly and falls on her knees to the ground.

This is one of the biggest revelations about Hawkeye we've witnessed. Roy is actually such a vital part of her life, that the sudden thought of losing him, makes her abandon all hope and lose her will to live completely. This is by no means a sign that Hawkeye is weak or incompetent. She just wasn't prepared for the shock of it. I'd say she's never wanted to deal with the possibility of Roy's death and thus her reaction took her by surprise.

The pain, the hollowness she felt the moment she faced the idea of Roy dying, made her unable to make straight thoughts. If she had been really pondering the situation, she would have noticed that Lust may have been lying... or mistaken.

Which was actually the case, and which she discovers when Roy arrives at the scene ready to erase Lust off the map. Roy's first reaction after all has ended, is to express his relief to see that she's all right and to thank Alphonse for protecting Hawkeye when she was unable to do it herself. Roy's too exhausted to keep his masks up and Hawkeye is just too shaken by the experience to even think of scolding him again.

The aftermath of this is a bit hard for both Roy and Hawkeye, as her actions were naive and not soldier-like at all. Now is Roy the one to scold her, much as it pains him. But they deal with it and keep going. Of course no one's gonna forget about what happened, but these kind of incidents (even if life-threatening and horrible) only brings them closer and adds even more valuable baggage to their relationship.