Everyone may think that there isn't anything that scares Natasha. Since she was young, she had seen every single type of horror life could give her, and she survived it. Everyone believes that Natasha is nearly invincible, and she doesn't need anything in order to survive. She does not even flinch when Tony tries to scare her, which had became a goal he tried to achieve everyday. She would just look at him with an amused smile and walks away while everyone laughs at Tony's feeble attempts.
She lets them believe whatever they want, but it doesn't fool Bruce.
He doesn't know what exactly her fear is, but he knows when she is facing them: at night.
Everyone isn't really shocked when they see Bruce and Natasha walk together for breakfast, even though Bruce hasn't moved into Natasha's apartment. They just assume the natural thing and go on with their life.
What they don't know is that Bruce slips into Natasha's bed and simply holds her when she is whimpering and sweating. He knows what it feels like it to be completely vulnerable in nightmares and waking up completely scared and horrified with what the mind comes up with.
No one isn't scared of nightmares. Even Natasha.
So he holds her until she relaxes and her breathing becomes normal. He too falls asleep with her by holding hands. When they wake up, they never talk about it. Bruce doesn't push it because he knows that if it was him then he wouldn't talked about it. So instead he just lets her know that he is here, whenever she wants or needs him. They smile to each other after a soft kiss, and get back to their life.
Yes, Natasha doesn't need anybody in her life to survive, but she wants Bruce to live her life fully.
There is a difference between surviving and living.