Disclaimer: I do not own the characters. I just play around with them too much for my own good.
Author's Note: This has much to do with the episodes "The Silence in the Libary"/"Forest of the Dead". I hope you enjoy!
He does not trust her because he does not know her, yet she claims to know him. He is not aware of ever having met her.
The Doctor does not like not knowing things.
This River Song seems to know too much for her own good.
He cannot trust her, even as possible death lurks around every corner and they dive deeper into a mystery that seems to bring up more questions than answers. He does not know what to make of her.
The Doctor fears the unknown.
She wants to prove herself, though. She insists upon it, so she leans forward and whispers the one thing he swore he would never tell anyone. His name.
His name. It is the very essence of who he is, the one constant in a life full of changes. It is his anchor.
In saying his name, she touches him deeper than anyone else ever has. With her whisper comes a great sense of fear and dread because someone else knows, and why would he show his true self completely to someone he doesn't even know yet?
To someone he will lose in the end.
The Doctor's name is his biggest secret. It is his true self, holding his darkest fears, deepest hatred, greatest sadness, and most powerful love. It is the one weapon that could destroy him should anyone find out about this, for if they knew his name they would know everything. It is something the Doctor cannot afford.
Yet she knows, this complete stranger who says she will be closer to him than anyone has been, who says that he is incomplete as of yet. She knows things he doesn't and it scares him.
The Doctor fears being exposed. He fears attachments. He fears the future, because he now knows that he will give himself completely to someone else, and the idea of that terrifies him more than anything.
Yet, for all the time travelling the Doctor has done, he knows he will not try and take away his name from River Song's memory.
The Doctor does all things for a good reason.
River Song knows his name, so he has no choice but to trust her.
And above all things, he trusts himself.